# A Topological Approach to Simple Descriptions of Convex Hulls of Sets
Defined by Three Quadrics

Grigoriy Blekherman and Alex Dunbar

###### Abstract.

We study the convex hull of a set $S\subset\mathbb{R}^{n}$ defined by three
quadratic inequalities. A simple way of generating inequalities valid on $S$
is to take nonnegative linear combinations of the defining inequalities of
$S$. We call such inequalities _aggregations_. We introduce a new technique
relating aggregations to properties of the spectral curve, i.e. the curve
defined by the vanishing of the determinant polynomial, and utilizing known
spectral sequences [1]. We find new families beyond those identified in [3,
6], where the convex hull is defined by aggregations. We also prove a
characterization of the emptiness of the projective variety defined by $3$
homogeneous quadratics in terms of the spectral curve generalizing results of
[2].

###### Key words and phrases:

convex hull, quadratic inequalities, spectral sequences, hyperbolic curves

###### 1991 Mathematics Subject Classification:

14P10, 52A20, 55T99

##  1\. Introduction

Optimization of linear functions on a subset $S$ of $\mathbb{R}^{n}$ leads
naturally to considering the convex hull of $S$. A semialgebraic set defined
by quadratic inequalities is often nonconvex. One strategy for computing the
convex hull of such a set is to study _aggregations_ of the defining
inequalities [3, 6, 17]. We focus on the case of three quadratic inequalities.
Given three linearly independent symmetric matrices
$Q_{1},Q_{2},Q_{3}\in\mathbb{R}^{(n+1)\times(n+1)}$ defining the three
quadratics
$f_{i}(x)=\begin{bmatrix}x^{\top}&1\end{bmatrix}Q_{i}\begin{bmatrix}x^{\top}&1\end{bmatrix}^{\top}$,
an aggregation is simply a nonnegative combination of the defining
inequalities: $f_{\lambda}=\sum_{i=1}^{3}\lambda_{i}f_{i}$ with
$\lambda_{i}\geq 0$. Note that $Q_{\lambda}=\sum_{i=1}^{3}\lambda_{i}Q_{i}$ is
a symmetric matrix which represents the quadratic function $f_{\lambda}$.

The use of aggregations for obtaining the convex hull of a set is a natural
idea in optimization. In linear programming, the Farkas Lemma ensures that the
convex hull of a set defined by linear inequalities can be described by
aggregations of these inequalities. Similarly, aggregations have been used in
the contexts of integer programming [4] and mixed-integer nonlinear
programming problems [8].

Let $S=\\{x\in\mathbb{R}^{n}\;|\;f_{i}(x)\leq 0,\;i\in[3]\\}$. One application
of the study of aggregations is to compute a representation of the closed
convex hull of $S$ as $\mathop{\overline{\rm
conv}}(S)=\bigcap_{\lambda\in\Lambda_{1}}\\{x\in\mathbb{R}^{n}\;|\;f_{\lambda}(x)\leq
0\\}$ for some appropriate subset $\Lambda_{1}$ of the nonnegative orthant
$\mathbb{R}^{3}_{+}$. In order to relate aggregations to convexity, we
restrict the signature of the matrices $Q_{\lambda}$. An aggregation $\lambda$
is _permissible_ if $Q_{\lambda}$ has at most one negative eigenvalue. If
$\lambda$ is a permissible aggregation, then the set
$\\{x\in\mathbb{R}^{n}\;|\;f_{\lambda}(x)<0\\}$ is either empty, convex, or
the union of two connected components, each of which is convex. A permissible
aggregation $\lambda$ is a _good_ aggregation if $\mathop{\overline{\rm
conv}}(S)\subseteq\\{x\in\mathbb{R}^{n}\;|\;f_{\lambda}(x)\leq 0\\}$. In order
to describe $\mathop{\overline{\rm conv}}(S)$ as an intersection of
permissible aggregations, it is necessary to further restrict our attention to
good aggregations. In [6], the authors prove such a description is possible
for sets defined by strict inequalities (i.e. $f_{i}(x)<0$ for $i\in[3]$)
under the assumption that there is a positive definite linear combination of
the defining quadratics. When such a linear combination exists, we say that
_PDLC_ holds. In [3], the authors show that for a set defined by (possibly
more than three, strict) quadratic inequalities, the convex hull can be
computed via aggregations under a hidden convexity assumption.

We study sets of aggregations of quadratic inequalities via the _spectral
curve_ , the curve in $\mathbb{RP}^{2}$ defined as the vanishing set of the
determinant
$g(\lambda)=\det(\lambda_{1}Q_{1}+\lambda_{2}Q_{2}+\lambda_{3}Q_{3})$, using
tools derived from algebraic topology. A line of work [1, 2, 5, 10] relates
the topology of sets defined by quadratic inequalities with the topology of
the sets of linear combinations of the $Q_{i}$ with specified number of
positive eigenvalues. In this setting, the homology groups of the solution
sets to systems of quadratic inequalities are computed from the cohomology
groups of combinations of the $Q_{i}$ with specified numbers of positive
eigenvalues using a spectral sequence. Note that the number of positive
eigenvalues is constant on connected components of the complement of the
spectral curve.

When PDLC holds, the projective variety
$\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})\subseteq\mathbb{RP}^{n}$
defined by the homogenizations of the $f_{i}$ is empty. We begin by
investigating the structure of the spectral curve for the case of three
quadratics which define an empty projective variety but do not necessarily
satisfy PDLC. Our first result shows that when $n\geq 4$ and the projective
variety $\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})$ is empty,
$g$ is _hyperbolic_ when it is smooth. Recall that a homogeneous polynomial
$h\in\mathbb{R}[x_{1},\ldots,x_{n+1}]$ is hyperbolic with respect to a point
$e$ if $h(te+a)\in\mathbb{R}[t]$ is real rooted for any point
$a\in\mathbb{R}^{n+1}$. If $h$ is a homogeneous polynomial in $n+1$ variables
which is hyperbolic with respect to a point $e$, then the connected component
of $e$ in $\mathbb{R}^{n+1}\setminus\mathcal{V}_{\mathbb{R}}(h)$ is a convex
cone [7]. If $h$ is hyperbolic and defines a smooth plane curve $C$, then $C$
consists of $\lfloor\frac{\deg h}{2}\rfloor$ nested ovals (and a pseudo-line
if $\deg h$ is odd). Theorem 1.1 is a generalization of the sufficient
condition that there is a positive definite linear combination $Q_{\mu}$ of
the $Q_{i}$ since in this case, $g$ is hyperbolic with respect to $\mu$.

######  Theorem 1.1.

Let $Q_{1},Q_{2},Q_{3}\in\mathbb{R}^{(n+1)\times(n+1)}$ be three linearly
independent symmetric matrices and $f_{1}^{h},f_{2}^{h},f_{3}^{h}$ the
associated quadratic forms $f_{i}^{h}(x)=x^{\top}Q_{i}x$. Suppose that the
spectral curve $\mathcal{V}_{\mathbb{R}}(g)$ is smooth and $n\not=2$. Then,
the real projective variety
$\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})$ is empty if and only
if one of the following holds:

  1. (i)

The polynomial $g$ is hyperbolic, and there is $\mu\in\mathbb{R}^{3}$ such
that $Q_{\mu}$ has $n$ positive eigenvalues

  2. (ii)

$n=3$ and the spectral curve $\mathcal{V}_{\mathbb{R}}(g)$ is empty.

When $n=2$, the spectral curve is not smooth if the variety
$\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})$ is nonempty.

The $n=3$ statement is [13, Theorem 7.8] and the statement that $g$ is
hyperbolic when $n\geq 4$ and the spectral curve is smooth is [2, Corollary
2]. However, we were unable to find a full statement of Theorem 1.1 in the
literature. Theorem 1.1 is proved in Section 5.

We observe that condition (i) in Theorem 1.1 gives the possible signatures
(number of positive eigenvalues minus number of negative eigenvalues) of the
matrices on the interior of the nested ovals of the spectral curve. On the
exterior of all ovals, the matrices must have signature 0 if $n+1$ is even and
$\pm 1$ if $n+1$ is odd [16]. Crossing from the exterior to the interior of an
oval makes the signature change by 2 so that matrices on the interior of the
second innermost oval (the oval of depth $\lfloor\frac{n+1}{2}\rfloor-1$) and
the exterior of the innermost convex oval have signature $\pm(n-1)$.
Therefore, there is a hyperbolicity cone of $g$ consisting of matrices which
are either positive definite or have $n-1$ positive and two negative
eigenvalues. Theorem 1.1 also has an interpretation in terms of the celebrated
Helton-Vinnikov Theorem [9], which shows that every hyperbolic plane curve
admits a definite determinantal representation (i.e. the defining quadratics
can be chosen to satisfy PDLC). In this case, the associated projective
variety is empty. Theorem 1.1 limits the other possible representations of
hyperbolic plane curves which correspond to empty varieties. Explicitly, the
only representations that correspond to an empty variety are definite
representations and representations for which $n$ positive eigenvalues are
attained and there is a hyperbolicity cone for which the matrices have $n-1$
positive and two negative eigenvalues. Moreover, such a representation
certifies the emptiness of the variety defined by the quadratics.

In the case that the real projective variety
$\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})$ is empty and
$n\not=2$, the spectral sequence approach relates certificates of emptiness
for the solution sets of systems of inequalities coming from aggregations to
the (non)intersections of polyhedral cones with a hyperbolicity cone of $g$.
In particular, computations of cohomology groups reduce to computations in
convex geometry.

Our primary application of such certificates of emptiness is a new sufficient
condition for the existence of a representation of $\mathop{\overline{\rm
conv}}(S)$ in terms of good aggregations. In [3, 6], hidden convexity
properties of the quadratic maps are leveraged to generate aggregations
$\lambda$ such that restriction $Q_{\lambda}|_{H}$ of $Q_{\lambda}$ to some
hyperplane $H\subseteq\mathbb{R}^{n+1}$ is positive semidefinite. When applied
to hyperplanes $H$ arising as the homogenizations of hyperplanes separating a
point $y\in\mathbb{R}^{n}$ from $\mathop{\rm conv}(\mathrm{int}(S))$, this
provides a good aggregation which certifies that $y\not\in\mathop{\rm
conv}(\mathrm{int}(S))$. Utilizing a spectral spectral sequence given in [1]
to compute homology groups of hyperplane sections of the solution set of a
system of quadratic inequalities, we employ a similar strategy and generalize
[6, Theorem 2.7] under the additional assumption that $S$ _has no points at
infinity_ , that is
$\left\\{x\in\mathbb{R}^{n}\;\middle|\;\begin{bmatrix}x^{\top}&0\end{bmatrix}Q_{i}\begin{bmatrix}x&0\end{bmatrix}^{\top}\leq
0\right\\}=\\{0\\}$.

######  Theorem 1.2.

Assume that $\mathrm{int}(S)\not=\emptyset$, and that
$\\{x\in\mathbb{R}^{n+1}\;|\;x^{\top}Q_{i}x\leq
0,\,i\in[3],\,x_{n+1}=0\\}=\\{0\\}$. Suppose that
$\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})$ is empty and the
spectral curve is smooth and nonempty.

  * (i)

If $n\geq 3$, then the spectral curve $g$ is hyperbolic. Let
$\mathcal{P}\subset\mathbb{R}^{3}$ be the hyperbolicity cone of $g$ such that
$\mathrm{int}\mathcal{P}$ consists of either positive semidefinite matrices
$Q_{\lambda}$, or matrices $Q_{\lambda}$ with exactly two negative
eigenvalues. If no non-zero aggregation lies in $\mathcal{P}$ then there are
$k\leq 6$ good aggregations
$\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(k)}$, such that
$\mathop{\overline{\rm
conv}}(S)=\\{x\in\mathbb{R}^{n}\;|\;f_{\lambda^{(i)}}(x)\leq 0,\,i\in[k]\\}.$

  * (ii)

If $n=2$ and $g$ is hyperbolic, then (i) still applies. If $g$ is not
hyperbolic, then there is a (possibly infinite) subset
$\Lambda_{1}\subseteq\mathbb{R}^{n}_{+}$ of good aggregations such that
$\mathop{\overline{\rm conv}}(S)=\\{x\in\mathbb{R}^{n}\;|\;f_{\lambda}(x)\leq
0,\,\lambda\in\Lambda_{1}\\}$.

Theorem 1.2 relaxes the PDLC condition from [6] but imposes the additional
restrictions that $S$ has no points at infinity and that the spectral curve is
smooth. However, the most complete results in the literature apply to the
setting of strict inequalities, while Theorem 1.2 applies to the (technically
more intricate) setting of non-strict inequalities. Theorem 1.2 is proved in
Section 7.

Our second application of certificates of emptiness is the development of a
strategy to prove the redundancy of an aggregation in a set, and in particular
reduce sets of aggregations to finite subsets. Specifically, given
aggregations $\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(k+1)}$, we
interpret the condition
$\bigcap_{i=1}^{k+1}S_{\lambda^{(i)}}=\bigcap_{i=1}^{k}S_{\lambda^{(i)}}$ in
terms of a system of equations and inequalities. An application of this
strategy shows that if the variety
$\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})$ is empty and the
spectral curve is smooth and hyperbolic, then the set obtained by intersecting
all permissible aggregations can be recovered by intersecting only finitely
many permissible aggregations.

######  Theorem 1.3.

Suppose that $\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})$ is
empty and that the spectral curve is smooth and hyperbolic. Then, there is a
finite subset $\Lambda_{1}$ of permissible aggregations such that

$\left\\{x\in\mathbb{R}^{n}\;|\;f_{\lambda}(x)\leq 0\text{ for all
}\lambda\in\Lambda_{1}\right\\}=\left\\{x\in\mathbb{R}^{n}\;|\;f_{\lambda}(x)\leq
0\text{ for all permissible }\lambda\in\mathbb{R}^{3}_{+}\right\\}.$

Theorem 1.3 is proved in Section 3.3.

Our final consideration is the topology of the set of good aggregations as a
subset of all permissible aggregations. Specifically, in Section 8, we
consider the connectedness properties of the set of good aggregations under
PDLC and the regularity assumption that $S$ has _no low dimensional
components_ , that is $S=\mathrm{cl}(\mathrm{int}(S))$. This leads us to a
bound on the number of good aggregations needed to describe
$\mathop{\overline{\rm conv}}(S)$ with these assumptions.

######  Theorem 1.4.

If $Q_{1},Q_{2},Q_{3}$ satisfy PDLC, $S=\mathrm{cl}(\mathrm{int}(S))$, $S$ has
no points at infinity, and $\mathrm{int}(S)\not=\emptyset$, then there is a
subset $\\{\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(r)}\\}$ of good
aggregations with $r\leq 4$ such that

$\mathop{\overline{\rm
conv}}(S)=\bigcap_{i=1}^{r}\left\\{x\in\mathbb{R}^{n}\;|\;f_{\lambda^{(i)}}(x)\leq
0\right\\}.$

Note that Theorem 1.4 is related to [3, Conjecture 3.2], in which the authors
conjecture the existence of a set $T$ defined by the intersection of three
_strict_ quadratic inequalities satisfying PDLC such that the convex hull
cannot be described using fewer than six good aggregations. The result of
Theorem 1.4 is that if the set defined by non-strict inequalities is
sufficiently regular, then four aggregations suffice. Theorem 1.4 is proved in
Section 3.4.

The remainder of the paper is organized as follows. In Section 2, we fix
notation and review the relevant background from real algebraic geometry and
algebraic topology. A detailed discussion of our main results and examples are
presented in Section 3. Section 4 presents conclusions. Proofs of the main
results are in subsequent sections.

##  2\. Notation and Preliminaries

We adopt the following notational conventions. For a positive integer $k$, the
set $\\{1,2,\ldots,k\\}$ is denoted $[k]$. Given a subset
$W\subseteq\mathbb{R}^{n}$, we let $\mathop{\rm conv}(W),\mathop{\rm
cone}(W),\mathrm{int}(W),\mathrm{cl}(W),$ and $\partial W$ be the convex hull,
conical hull, interior, closure, and boundary of $W$, respectively. We denote
by $\mathop{\overline{\rm conv}}(W)$ and $\mathop{\overline{\rm cone}}(W)$ the
closures of the convex hull and conical hull of $W$, respectively. The real
projective $n$-space is denoted $\mathbb{RP}^{n}$, the $n$-sphere is denoted
$\mathbb{S}^{n}$, the closed unit $n$-ball is denoted $B^{n}$, and the
nonnegative orthant of $\mathbb{R}^{n}$ is denoted $\mathbb{R}^{n}_{+}$. If
$\lambda\in\mathbb{R}^{n}$ is a vector, then $\lambda_{i}$ denotes the
$i^{th}$ component of $\lambda$ and
$\operatorname*{supp}(\lambda)=\\{i\in[n]\;|\;\lambda_{i}\not=0\\}$ is the
support of $\lambda$. The standard basis vectors of $\mathbb{R}^{n}$ are
denoted by $e_{i}$, where $(e_{i})_{j}=\begin{cases}1&i=j\\\
0&\text{otherwise}\end{cases}$.

Throughout, we will consider the setting with three fixed symmetric matrices
$Q_{1},Q_{2},Q_{3}\in\mathbb{R}^{(n+1)\times(n+1)}$. For each $i\in[3]$, we
have the block structure $Q_{i}=\begin{bmatrix}A_{i}&b_{i}\\\
b_{i}^{\top}&c_{i}\end{bmatrix}$, where $A_{i}\in\mathbb{R}^{n\times
n},b_{i}\in\mathbb{R}^{n},$ and $c_{i}\in\mathbb{R}$. To these three matrices
we associate the following:

  * •

The quadratic functions
$f_{i}(x)=\begin{bmatrix}x^{\top}&1\end{bmatrix}Q_{i}\begin{bmatrix}x^{\top}&1\end{bmatrix}^{\top}$,
$i\in[3]$.

  * •

The set $S=\\{x\in\mathbb{R}^{n}\;|\;f_{i}(x)\leq 0,i\in[3]\\}$

  * •

The homogenized functions
$f_{i}^{h}(x,x_{n+1})=\begin{bmatrix}x^{\top}&x_{n+1}\end{bmatrix}Q_{i}\begin{bmatrix}x^{\top}&x_{n+1}\end{bmatrix}^{\top}$,
$i\in[3]$.

  * •

The homogenized set $S^{h}=\\{(x,x_{n+1})\in\mathbb{R}^{n+1}\;|\;f_{i}(x)\leq
0,i\in[3]\\}$.

  * •

The determinant polynomial $g(x,y,z)=\det(xQ_{1}+yQ_{2}+zQ_{3})$ and the
spectral curve $\mathcal{V}_{\mathbb{R}}(g)\subseteq\mathbb{RP}^{2}$.

Moreover, given $\lambda\in\mathbb{R}^{3}$, we consider linear combinations:

  * •

$Q_{\lambda}=\sum_{i=1}^{3}\lambda_{i}Q_{i}$. The blocks
$A_{\lambda},b_{\lambda},$ and $c_{\lambda}$ are defined analogously.

  * •

$f_{\lambda}(x)=\sum_{i=1}^{3}\lambda_{i}f_{i}(x)$ and the homogenization
$f_{\lambda}^{h}=\sum_{i=1}^{3}\lambda_{i}f_{i}^{h}(x)$.

  * •

$S_{\lambda}=\\{x\in\mathbb{R}^{n}\;|\;f_{\lambda}(x)\leq 0\\}$ and its
homogenization
$S_{\lambda}^{h}=\\{(x,x_{n+1})\in\mathbb{R}^{n+1}\;|\;f_{\lambda}^{h}(x)\leq
0\\}$.

We set

$\Lambda=\\{\lambda\in\mathbb{R}^{3}_{+}\;|\;Q_{\lambda}\text{ has exactly one
negative eigenvalue}\\}.$

Note that $\lambda$ is a permissible aggregation if and only if
$\lambda\in\Lambda$ or $Q_{\lambda}\succeq 0$. Moreover, if $Q_{\lambda}$ has
exactly one negative eigenvalue, then $\text{int}(S_{\lambda})$ consists of
either one convex component or two connected components, each of which is
convex [3, Lemma 5.2]. It is possible that $\lambda\in\Lambda$ and
$\mathrm{int}(S_{\lambda})$ has two connected components, each of which
intersects $\mathrm{int}(S)$ nontrivially. In this case, $\mathop{\rm
conv}(S)\not\subseteq S_{\lambda}$. So, we additionally define the set of
_good aggregations_ to be the $\lambda\in\Lambda$ such that $\mathop{\rm
conv}(S)\subseteq S_{\lambda}$.

Let $K\subseteq\mathbb{R}^{m}$ be a polyhedral cone. We denote the polar dual
by $K^{\circ}=\\{x\in\mathbb{R}^{m}\;|\;x^{\top}y\leq 0\text{ for all }y\in
K\\}$. For a homogeneous quadratic map
$f^{h}=(f_{1}^{h},f_{2}^{h},\ldots,f_{m}^{h}):\mathbb{R}^{n+1}\to\mathbb{R}^{m}$,
let

$X(K,f^{h})=\\{[x]\in\mathbb{RP}^{n}\;|\;f^{h}([x])\in K\\}.$

In particular, given vectors
$\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(m)}\in\mathbb{R}^{3}_{+}$, the
homogeneous quadratic map
$f^{h}=(f_{\lambda^{(1)}}^{h},f_{\lambda^{(2)}}^{h},\ldots,f_{\lambda^{(m)}}^{h})$,
and the cone $K=-\mathbb{R}^{m}_{+}$, we have that

$X(-\mathbb{R}^{m}_{+},f^{h})=\left\\{[x]\in\mathbb{RP}^{n}\;\middle|\;x\in\bigcap_{i=1}^{m}S_{\lambda^{(i)}}^{h},x\not=0\right\\}.$

Note that the condition $f^{h}([x])\in K$ is well-defined because $f^{h}$ is
homogeneous quadratic and $K$ is a cone and therefore closed under
multiplication by positive scalars. Finally, let
$\Omega(K)=K^{\circ}\cap\mathbb{S}^{m-1}$, $C\Omega(K)=K^{\circ}\cap B^{m}$,
and

$\Omega^{j}(K)=\\{\lambda\in\Omega(K)\;|\;Q_{\lambda}\text{ has at least
}j\text{ positive eigenvalues}\\}.$

We will primarily work with the cases $K=\\{0\\}$ and $K=-\mathbb{R}^{3}_{+}$.
When $K=\\{0\\}$ and $K^{\circ}=\mathbb{R}^{3}$, the set $X(K,f^{h})$ is the
projective variety
$\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})\subseteq\mathbb{RP}^{n}$
and the sets $\Omega^{j}(K)$ contain the (normalized) coefficients of linear
combinations of the matrices $Q_{i}$ with at least $j$ positive eigenvalues.
In the case $K=-\mathbb{R}^{3}_{+}$ and $K^{\circ}=\mathbb{R}^{3}_{+}$,
$X(K,f^{h})$ is the solution set of the system of inequalities
$f_{i}^{h}([x])\leq 0$ for $i=1,2,3$ and the sets $\Omega^{j}(K)$ contain the
(normalized) aggregations corresponding to matrices with at least $j$ positive
eigenvalues. We omit $K$ from the notation and write e.g. $\Omega(K)=\Omega$
and $X(K,f^{h})=X$ when the cone $K$ and homogeneous quadratic map $f^{h}$ are
clear from the context.

We frequently work with homogeneous quadratic maps defined by aggregations
$\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(m)}\in\Lambda$. In this setting,
if $A\in\mathbb{R}^{m\times 3}$ is the matrix whose rows are
$(\lambda^{(j)})^{\top}$, then the homogeneous quadratic map
$f_{A}^{h}=(f_{\lambda^{(1)}}^{h},\ldots,f_{\lambda^{(m)}}^{h}):\mathbb{R}^{n+1}\to\mathbb{R}^{m}$
factors though $\mathbb{R}^{3}$ as

${\mathbb{R}^{n+1}}$${\mathbb{R}^{3}}$${\mathbb{R}^{m}}$${x}$${f^{h}(x)=\begin{bmatrix}x^{\top}Q_{1}x\\\
x^{\top}Q_{2}x\\\
x^{\top}Q_{3}x\end{bmatrix}}$${Af^{h}(x)=\begin{bmatrix}f_{\lambda^{(1)}}^{h}(x)\\\
\vdots\\\ f_{\lambda^{(m)}}^{h}(x)\end{bmatrix}}$$\scriptstyle{A}$

where
$f^{h}=(f_{1}^{h},f_{2}^{h},f_{3}^{h}):\mathbb{R}^{n+1}\to\mathbb{R}^{3}$ is
the homogeneous quadratic map defined by the $Q_{i}$.

Now, for a fixed polyhedral cone $K\subseteq\mathbb{R}^{k+1}$, the set
$X(K,f_{A}^{h})=\\{[x]\in\mathbb{RP}^{n}\;|\;f_{A}^{h}(x)\in K\\}$ is the same
as the set $X(A^{-1}(K),f^{h})=\\{[x]\in\mathbb{RP}^{n}\;|\;f^{h}(x)\in
A^{-1}(K)\\}$, where $A^{-1}(K)=\\{v\in\mathbb{R}^{3}\;|\;Av\in K\\}$ is the
preimage of $K$ under the map defined by $A$. Computing polar duals gives that

$\displaystyle(A^{\top}K^{\circ})^{\circ}$
$\displaystyle=\\{v\in\mathbb{R}^{3}\;|\;\langle A^{\top}y,v\rangle\leq
0\,\text{ for all }y\in K^{\circ}\\}$
$\displaystyle=\\{v\in\mathbb{R}^{3}\;|\;\langle y,Av\rangle\leq 0\,\text{ for
all }y\in K^{\circ}\\}$
$\displaystyle=\\{v\in\mathbb{R}^{3}\;|\;Av\in(K^{\circ})^{\circ}\\}$
$\displaystyle=A^{-1}(K).$

So, $A^{-1}(K)$ has dual cone $A^{\top}K^{\circ}\subseteq\mathbb{R}^{3}$. To
compute homology groups of $X(K,f_{A}^{h})$ using Theorem 3.1, it therefore
suffices to consider dual cones in $\mathbb{R}^{3}$ of the form
$A^{\top}K^{\circ}$. In particular, we have

$X(-\mathbb{R}^{m}_{+},f_{A}^{h})=X(\mathop{\rm
cone}(\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(m)})^{\circ},f^{h}).$

Given a hyperplane $H\subseteq\mathbb{R}^{n+1}$, we denote by
$Q_{\lambda}|_{H}$ a matrix representative of the restriction of the quadratic
form $f^{h}_{\lambda}$ to $H$. In particular, if $B\in\mathbb{R}^{(n+1)\times
n}$ is a matrix whose columns form an orthonormal basis of $H$ then a matrix
representative is given by $B^{\top}Q_{\lambda}B$. The signature and
singularity of the matrices $Q_{\lambda}|_{H}$ is independent of the choice of
basis of $H$. We denote

$\Omega^{j}_{H}(K)=\left\\{\lambda\in\Omega(K)\;\middle|\;Q_{\lambda}|_{H}\text{
has at least }j\text{ positive eigenvalues}\right\\}$

for $j=0,1,\ldots,n$. Throughout, (co)homology groups are taken with
$\mathbb{Z}_{2}$ coefficients. It is a central result of [1] that homology
groups of $X(K,f^{h})$ can be computed from the relative cohomology groups of
the pairs $(C\Omega(K),\Omega^{j}(K))$.

###  2.1. Background

In this section we recall relevant properties of real algebraic plane curves,
computations with spectral sequences, and permissible aggregations.

####  2.1.1. Real Algebraic Plane Curves

Let $h\in\mathbb{R}[x,y,z]$ be a homogeneous polynomial of degree $d$. The
variety $C=\mathcal{V}_{\mathbb{R}}(h)\subseteq\mathbb{RP}^{2}$ is a _real
algebraic plane curve_. A nonsingular real algebraic plane curve $C$ can have
multiple connected components. If a connected component $C_{1}$ of $C$ is
contractible in $\mathbb{RP}^{2}$ and $\mathbb{RP}^{2}\setminus C_{1}$ has two
connected components, exactly one of which is contractible, then $C_{1}$ is
called an _oval_. The contractible connected component of
$\mathbb{RP}^{2}\setminus C_{1}$ is called the _interior_ of $C_{1}$ and the
non-contractible component of $\mathbb{RP}^{2}\setminus C_{1}$ is the
_exterior_. If $d$ is even, then every component of $C$ is an oval. If $d$ is
odd, then there is an additional non-oval component. If
$C_{1},C_{2},\ldots,C_{k}$ are ovals such that $C_{i}$ is contained in the
interior of $C_{j}$ for $1\leq j<i\leq k$ and are the maximal set of ovals
satisfying this property, then $C_{k}$ is said to be an oval of _depth $k$_
and the set $C_{1},C_{2},\ldots,C_{k}$ is a _nest_ of ovals. The paper [5]
gives a thorough account of possible arrangements of ovals.

We are particularly interested in the case that there is a nest of ovals in
$C$ of maximal depth. This maximal nest must have depth
$\lfloor\frac{d}{2}\rfloor$. In this case, the curve $C$ is the hypersurface
of a _hyperbolic_ polynomial $h\in\mathbb{R}[x,y,z]$. Recall that a
homogeneous polynomial $h\in\mathbb{R}[x,y,z]$ is called hyperbolic with
respect to $e\in\mathbb{R}^{3}$ if the univariate polynomial
$h(te+a)\in\mathbb{R}[t]$ is real rooted for any $a\in\mathbb{R}^{3}$. The
connected component of $e$ in
$\mathbb{R}^{3}\setminus\mathcal{V}_{\mathbb{R}}(h)$ is an open convex cone
called a _hyperbolicity cone_ of $h$ and $h$ is hyperbolic with respect to any
point $e^{\prime}$ in this cone [7]. Note that if $\mathcal{P}$ is a
hyperbolicity cone of $h$, then its image in $\mathbb{RP}^{2}$ is the interior
of the oval of $C=\mathcal{V}_{\mathbb{R}}(h)$ of maximal depth. Every
hyperbolic plane curve possesses a definite determinantal representation [9],
but a determinantal representation of a hyperbolic plane curve is not
necessarily definite. See [12] for an account of determinantal representations
of hyperbolic plane curves.

####  2.1.2. Spectral Sequences

Here we provide a brief introduction to spectral sequences, focusing on the
application to computing homology groups rather than an account of the
underlying theory. A thorough introduction to spectral sequences is given in
[11].

A _(first quadrant cohomology) spectral sequence_ $(E_{r},d_{r})$ is a
collection of pages $E_{r}$ and differentials $d_{r}:E_{r}\to E_{r}$ for
$r\geq 0$. Each page $E_{r}$ consists of a collection of
$\mathbb{Z}_{2}$-vector spaces $E_{r}^{i,j}$ indexed by $\mathbb{Z}^{2}$,
where $E_{r}^{i,j}=0$ if $i<0$ or $j<0$. Each differential is a collection of
linear maps $d_{r}^{i,j}:E^{i,j}_{r}\to E^{i+r,j-r+1}_{r}$ such that
$d_{r}\circ d_{r}=0$. The $r+1$-st page of the spectral sequence has terms
given by the homology of $d_{r}$. Precisely,

$E_{r+1}^{i,j}=\frac{\ker(d_{r}^{i,j})}{\mathrm{im}(d_{r}^{i-r,j+r-1})}.$

Since $E_{r}^{i,j}=0$ when $i<0$ or $j<0$, it follows that for sufficiently
large $r$, $\ker(d_{r}^{i,j})=E^{i,j}_{r}$ and $\mathop{\rm
im}(d_{r}^{i-r,j+r-1})=0$ so that $E_{r+1}^{i,j}\simeq E_{r}^{i,j}$ and
$E_{k}^{i,j}\simeq E_{r}^{i,j}$ for all sufficiently large $k$. In this case,
we say that the $i,j$ term of the spectral sequence has _converged_ and write
$E_{r}^{i,j}=E_{k}^{i,j}=E_{\infty}^{i,j}$. We additionally say that
$(E_{r},d_{r})$ _converges_ to some graded $H_{*}$ if
$H_{*}\simeq\bigoplus_{i+j=*}E_{\infty}^{i,j}$.

######  Example 2.1.

We conclude this section with an example computation of the $E_{3}$ page of a
spectral sequence given the $E_{2}$ page. This computation forms the basis of
our proof of Theorem 3.3 below.

Suppose that $(E_{r},d_{r})$ is a first quadrant cohomology spectral sequence
which converges to $H_{n-*}$. Suppose further that
$E_{2}^{i,j}=\begin{cases}\mathbb{Z}_{2},&(i,j)=(0,n)\text{ or }(2,n-1)\\\
0,&\text{otherwise}\end{cases}$ and $d_{2}^{0,n}:E_{2}^{0,n}\to E_{2}^{2,n-1}$
is an isomorphism. The $E_{3}$ page then has
$E_{3}^{0,n}=\ker(d_{2}^{0,n})/\mathop{\rm im}(d_{2}^{-2,n+1})=0/0\simeq 0$
and $E_{3}^{2,n-1}=\ker(d_{2}^{2,n-1})/\mathop{\rm
im}(d_{2}^{0,n})=\mathbb{Z}_{2}/\mathbb{Z}_{2}\simeq 0$. So, the $E_{3}$ page
has $E_{3}^{i,j}=0$ for all $i,j$ and therefore $E_{\infty}^{i,j}=0$ for all
$i,j$. This is represented graphically by

[title = $E_{\page}$, no y ticks, name = Spectral Sequence example,
cohomological Serre grading, classes = draw = none ] [background] at (-1,0)
n-1; at (-1,1) n; [”Z_2”](0,1) [”0”](0,0) [”0”](1,1) [”0”](1,0) [”0”](2,1)
[”Z_2”](2,0) [̣”d_2^0,n” description]2(0,1) [”0”] [”0”]

$\printpage[name=SpectralSequenceexample,page=2]\qquad\qquad\printpage[name=SpectralSequenceexample,page=3]$

Since $E_{3}^{i,j}=0$ for all $i,j$, it follows that $E_{\infty}^{i,j}=0$ for
all $i,j$ and therefore we have that
$H_{0}\simeq\bigoplus_{i+j=n}E_{\infty}^{i,j}\simeq 0$.

####  2.1.3. Permissible Aggregations

We are interested in aggregations $\lambda\in\mathbb{R}^{3}_{+}$ where the
associated set $S_{\lambda}$ has convex components. This is reflected in the
signature of $Q_{\lambda}$ as follows.

######  Proposition 2.2 ([17, Proposition 4]).

Let $\lambda\in\mathbb{R}^{3}$ and suppose that
$\mathrm{int}(S_{\lambda}^{h})\not=\emptyset$. The following are equivalent:

  1. (1)

There is a linear hyperplane which does not intersect
$\mathrm{int}(S_{\lambda}^{h})$.

  2. (2)

$Q_{\lambda}$ has exactly one negative eigenvalue

  3. (3)

$\mathrm{int}(S_{\lambda}^{h})$ is a union of two disjoint open convex cones
which are symmetric reflections of each other with respect to the origin.

When the equivalent conditions of Proposition 2.2 are satisfied,
$\mathrm{int}(S_{\lambda}^{h})$ is called a _semi-convex cone_. The existence
of aggregations with exactly one negative eigenvalue is reflected in the
geometry of the set $S^{h}$ as follows.

######  Proposition 2.3.

If there exists $\lambda\in\mathbb{R}^{3}_{+}$ such that $Q_{\lambda}$ has $n$
positive and one negative eigenvalue, then there is a hyperplane $H$ such that
$S^{h}\cap H=0$.

Conversely, suppose that $S^{h}\cap H=0$ for some hyperplane $H$ such that the
$Q_{i}|_{H}$ satisfy PDLC. Then, there exists $\lambda\in\mathbb{R}^{3}_{+}$
such that $Q_{\lambda}$ has at most one negative eigenvalue.

###### Proof.

For the first statement, take $H$ to be the hyperplane spanned by the $n$
vectors corresponding to positive eigenvalues of $Q_{\lambda}$.

For the second statement, if $S^{h}\cap H=0$, then
$H\cap\mathrm{int}(S^{h})=\\{(x,x_{n+1})\in\mathbb{R}^{n+1}\cap
H\;|\;f^{h}(x,x_{n+1})<0\\}=\emptyset$. By the S-lemma variant [6, Lemma 2.3],
this implies the existence of $\lambda\in\mathbb{R}^{3}_{+}$ with
$Q_{\lambda}|_{H}\succeq 0$. So, $Q_{\lambda}$ has at most one negative
eigenvalue by the Cauchy Interlacing Theorem. ∎

We are further interested in aggregations $\lambda\in\Lambda$ where
$S_{\lambda}$ satisfies $\mathop{\rm conv}(S)\subseteq S_{\lambda}$. We call
such $\lambda$ _good aggregations_. An aggregation $\lambda\in\Lambda$ which
is not a good aggregation is a _bad aggregation_.

######  Lemma 2.4 ([3, Lemma 5.2]).

If $\lambda\in\Lambda$, then $\mathrm{int}(S_{\lambda})$ is either a convex
set or a union of two disjoint convex sets. Moreover, $\lambda$ is not a good
aggregation if and only if $\mathrm{int}(S_{\lambda})$ is a union of two
disjoint convex sets and $\mathrm{int}(S)$ has nonempty intersection with both
components.

Note that Lemma 2.4 implies that if the set $S$ is connected then every
$\lambda\in\Lambda$ is a good aggregation.

##  3\. Main Results

In this section we describe our main results in detail and provide examples.

###  3.1. Systems of Quadratics with No Solutions

Our first goal is to apply the results of [1] to certify nonexistence of
solutions to systems of quadratics. The primary tool is a spectral sequence
relating the homology groups of $X=X(K,f^{h})$ to the relative cohomology
groups of pairs $(C\Omega,\Omega^{j})$.

######  Theorem 3.1 ([1, Theorem A]).

There is a first quadrant cohomology spectral sequence $(E_{r},d_{r})$
converging to $H_{n-*}(X)$ such that $E_{2}^{ij}=H^{i}(C\Omega,\Omega^{j+1})$.

Note that the set $C\Omega$ is the intersection of a polyhedral cone and the
unit ball and is therefore contractible. So, from the long exact sequence for
relative cohomology,

$\dots\to H^{i}(C\Omega,\Omega^{j+1})\to H^{i}(C\Omega)\to
H^{i}(\Omega^{j+1})\to H^{i+1}(C\Omega,\Omega^{j+1})\to\dots,$

we see that

(1) $E_{2}^{ij}\cong\begin{cases}H^{i-1}(\Omega^{j+1}),&i\geq
2,\;\Omega^{j+1}\not=\emptyset\\\
H^{0}(\Omega^{j+1})/\mathbb{Z}_{2},&i=1,\;\Omega^{j+1}\not=\emptyset\\\
\mathbb{Z}_{2},&i=0,\;\Omega^{j+1}=\emptyset\\\
0,&\text{otherwise}\end{cases}.$

The differential $d_{2}$ has an explicit formula provided by [1, Theorem B].
Example 3.2 below demonstrates how the spectral sequence of Theorem 3.1 and
the isomorphisms (1) combine to relate the homology of a variety to the
cohomology groups of linear combinations of the defining quadratic forms with
specified signature.

######  Example 3.2.

This example is modified from [12, Example 5.2]. Consider the matrices

$Q_{1}=\begin{bmatrix}25&0&-32&0\\\ 0&25&0&24\\\ -32&0&6&0\\\
0&24&0&6\end{bmatrix},\quad Q_{2}=\begin{bmatrix}0&0&12&0\\\ 0&0&0&16\\\
12&0&4&0\\\
0&16&0&4\end{bmatrix},\;\text{and}\;Q_{3}=\begin{bmatrix}0&0&0&-60\\\
0&0&10&0\\\ 0&10&0&0\\\ -60&0&0&0\end{bmatrix}.$

It is shown in [12, Example 5.2] that
$g(\lambda_{1},\lambda_{2},\lambda_{3})=\det(\lambda_{1}Q_{1}+\lambda_{2}Q_{2}+\lambda_{3}Q_{3})$
is smooth and hyperbolic with respect to $(1,0,0)$ but PDLC is not satisfied.
For $K=\\{0\\}$, we compute that $\Omega^{1}(K)=\mathbb{S}^{2}$,
$\Omega^{2}(K)$ is homotopy equivalent to the union of two points, and
$\Omega^{3}(K)$ is homotopy equivalent to $\mathbb{S}^{1}$. So, the nonzero
entries on the $E_{2}$ page and the potentially nonzero entries on the $E_{3}$
page are given by

[title = $E_{\page}$, name = Hyperbolic example, cohomological Serre grading,
classes = draw = none , xscale = 2 ] [”Z_2”](0,3) [”Z_2”](1,1) [”Z_2”](2,2)
[”Z_2”](3,0)

[̣”d_2^0,3” description]2(0,3) [” ker(d_2^0,3)”] [”Z2im(d20,3)”] [̣”d_2^1,1”
description]2(1,1) [”ker(d_2^1,1)”] [”Z2im(d21,1)”]

$\printpage[name=Hyperbolicexample,page=2]\quad\printpage[name=Hyperbolicexample,page=3]$

Moreover, [12, Example 5.2] shows that if $f_{i}^{h}$ is the quadratic form
associated to $Q_{i}$ for $i=1,2,3$, then the variety
$\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})=\emptyset$. By the
computation of the $E_{3}$ page, and the fact that $d_{r}=0$ for $r\geq 3$, we
see that $d_{2}^{0,3}$ and $d_{2}^{1,1}$ are isomorphisms.

In the case that $K=\\{0\\}$ and
$X(\\{0\\},f^{h})=\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})\subseteq\mathbb{RP}^{n}$,
the emptiness of $X$ is certified by the hyperbolicity of the spectral curve
together with signature information.

######  Theorem 3.3.

The projective variety
$\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})\subseteq\mathbb{RP}^{n}$
is empty if $\Omega^{n}(\\{0\\})\not=\emptyset$ (i.e. there is
$\mu\in\mathbb{R}^{3}$ such that $Q_{\mu}$ has at least $n$ positive
eigenvalues) and $g$ is smooth and hyperbolic. The converse holds when $n=1$
or $n\geq 4$.

The statement that when $n\geq 4$, emptiness of $X(\\{0\\},f^{h})$ implies the
hyperbolicity of $g$ is [2, Corollary 2]. Note that the case
$\Omega^{n+1}\not=\emptyset$ implies that $Q_{1},Q_{2},Q_{3}$ satisfy PDLC,
which is considered in the context of aggregations of quadratic inequalities
in [6]. Additionally, Theorem 3.3 is related to the result [13, Theorem 7.8],
which characterizes the emptiness of
$X=\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})$ in the case $n=3$.
In this case, an empty spectral curve also certifies that $X=\emptyset$.

The characterization of the emptiness of projective varieties leads to a
computation for solution sets of inequalities.

######  Proposition 3.4.

If
$X(\\{0\\},f^{h})=\mathcal{V}_{\mathbb{R}}(f^{h}_{1},f^{h}_{2},f^{h}_{3})=\emptyset$,
then for a polyhedral cone $K\not=0$, the set $X(K,f^{h})=\emptyset$ if and
only if $\Omega^{n}(K)\not=\emptyset$ and either $H^{1}(\Omega^{n}(K))\neq 0$
or $\Omega^{n+1}(K)\not=\emptyset$.

The result of Proposition 3.4 is that non-existnce of solutions of systems of
$3$ quadratic inequalities can be certified via convex geometry when the
polynomial $g$ is smooth and hyperbolic. Indeed, $H^{1}(\Omega^{n}(K))\not=0$
means that there is a noncontractible loop in the set of combinations of the
matrices $Q_{i}$ with coefficients in the dual cone $K^{\circ}$ which have at
least $n$ positive eigenvalues. This occurs if and only if PDLC is not
satisfied and the hyperbolicity cone of $g$ corresponding to matrices with two
negative eigenvalues is contained in $K^{\circ}$, while
$\Omega^{n+1}(K)\not=\emptyset$ if and only if PDLC is satisfied and
$K^{\circ}$ intersects the cone of positive definite matrices. On the other
hand, in the exceptional cases $n=2,3$ where the variety is empty but $g$ is
not hyperbolic, any system of inequalities which is not equivalent to the
vanishing of all $f_{i}^{h}$ has a solution.

###  3.2. A Sufficient Condition for Obtaining the Convex Hull via
Aggregations

Here, we apply the topological machinery to develop a new sufficient condition
for $\mathop{\overline{\rm conv}}(S)$ to be given by good aggregations. In [3,
6], sufficient conditions for the convex hull to be given by aggregations are
derived using the following idea.

######  Proposition 3.5.

Suppose that $\mathrm{int}(S)\not=\emptyset$ and that the matrices $Q_{i}$
satisfy the following property:

(*) $\begin{split}&\text{ If }\alpha^{\top}x<\beta\text{ is a valid inequality
on }S,\text{ then for the hyperplane }\\\
&H=\\{(x,x_{n+1})\;|\;\alpha^{\top}x=\beta
x_{n+1}\\}\subseteq\mathbb{R}^{n+1},\text{ there exists
}\lambda\in\mathbb{R}^{3}_{+}\text{ such that }Q_{\lambda}|_{H}\succ
0.\end{split}$

Then, there is a set $\Lambda_{1}\subseteq\mathbb{R}^{3}_{+}$ of good
aggregations such that $\mathop{\overline{\rm
conv}}(S)=\bigcap_{\lambda\in\Lambda_{1}}S_{\lambda}$.

In [3, 6], condition (* ‣ 3.5) is imposed via hidden convexity results. In
particular, [6] requires PDLC and [3] requires hidden hyperplane convexity
(i.e. the restriction of $f^{h}$ to any hyperplane has convex image). Both
conditions imply property (* ‣ 3.5).

Proposition 3.4 implies that the two possibilities for certificates that
$S^{h}\cap H=0$ are the existence of a PSD aggregation of $Q_{i}|_{H}$ or a
noncontractible loop in the set of aggregations of the matrices $Q_{i}|_{H}$
with at least $n-1$ positive eigenvalues (i.e.,
$H^{1}(\Omega_{H}^{n-1}(-\mathbb{R}^{3}_{+}))\neq 0$). So, we seek a condition
under which every certificate takes the form of an aggregation $\lambda$ with
$Q_{\lambda}|_{H}\succeq 0$. When the polynomials
$g(\lambda)=\det(Q_{\lambda})$ and $g_{H}(\lambda)=\det(Q_{\lambda}|_{H})$ are
both hyperbolic, the first step towards such a condition is understanding the
relationship between the hyperbolicity cones $\mathcal{P}$ of $g$ and
$\mathcal{P}_{H}$ of $g_{H}$. If the matrices $Q_{i}$ satisfy PDLC, then
$g_{H}$ interlaces $g$ so that $\mathcal{P}\subseteq\mathcal{P}_{H}$. If PDLC
is not satisfied, the following relationship between $\mathcal{P}$ and
$\mathcal{P}_{H}$ holds.

######  Theorem 3.6.

Suppose $\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})=\emptyset$
and $g$ is smooth and hyperbolic with a hyperbolicity cone $\mathcal{P}$ which
contains matrices with exactly two negative eigenvalues. Then, if $g_{H}$ is
smooth and hyperbolic, either $\mathcal{P}_{H}$ contains positive definite
matrices or $\mathcal{P}_{H}\subseteq\mathcal{P}$.

Combining the ideas in Proposition 3.5 and Theorem 3.6 suggests that in order
to describe $\mathop{\overline{\rm conv}}(S)$ using good aggregations, it
suffices to separate the hyperbolicity cone $\mathcal{P}$ of $g$ from the cone
$\mathbb{R}_{+}^{3}$ of aggregations. We say that the set $S$ _has no points
at infinity_ if $\\{x\in\mathbb{R}^{n}\;|\;f_{i}^{h}(x,0)\leq 0\text{ for all
}i\in[3]\\}=\\{0\\}$.

######  Theorem 3.7.

Suppose that
$\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})=\emptyset$, the
polynomial $g$ is smooth and hyperbolic, $\mathrm{int}(S)\not=\emptyset$, and
$S$ has no points at infinity. Let $\mathcal{P}$ be the hyperbolicity cone of
$g$ where $\lambda\in\mathrm{int}\mathcal{P}$ implies that either
$Q_{\lambda}\succ 0$ or $Q_{\lambda}$ has exactly two negative eigenvalues. If
no nonzero aggregation lies in $\mathcal{P}$, then $\mathop{\overline{\rm
conv}}(S)=\bigcap_{\lambda\in\Lambda_{1}}S_{\lambda}$ for some set of good
aggregations $\Lambda_{1}$.

We demonstrate by an example that $\mathop{\overline{\rm conv}}(S)$ may not be
given by aggregations when a nonzero aggregation lies in $\mathcal{P}$.

######  Example 3.8.

This example is modified from [12, Example 5.2]. Consider the matrices

$Q_{1}=\begin{bmatrix}25&0&-32&0\\\ 0&25&0&24\\\ -32&0&6&0\\\
0&24&0&6\end{bmatrix},\quad Q_{2}=\begin{bmatrix}0&0&12&0\\\ 0&0&0&16\\\
12&0&4&0\\\
0&16&0&4\end{bmatrix},\;\text{and}\;Q_{3}=\begin{bmatrix}0&0&0&-60\\\
0&0&10&0\\\ 0&10&0&0\\\ -60&0&0&0\end{bmatrix}.$

It is shown in [12, Example 5.2] that
$g(\lambda_{1},\lambda_{2},\lambda_{3})=\det(\lambda_{1}Q_{1}+\lambda_{2}Q_{2}+\lambda_{3}Q_{3})$
is hyperbolic with respect to $(1,0,0)$ and that the real variety
$\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})\subseteq\mathbb{RP}^{3}$
is empty. Here the convex hull of $S$ is not given by aggregations, as shown
in Figure 1.

Figure 1. The spectral curve and its relation to the defining inequalities in
Example 3.8 (left). Note that one of the defining inequalities is contained in
the hyperbolicity cone of $g$ and that the intersection of $S_{\lambda}$ for
$\lambda\in\Lambda^{\prime}$ gives a strict superset of $\mathop{\overline{\rm
conv}}(S)$ (right).

On the other hand, if we modify $f_{1}$ to be

(2)
$\tilde{f}_{1}=\begin{bmatrix}x^{\top}&1\end{bmatrix}\left(0.3Q_{1}+0.4Q_{2}+0.3Q_{3}\right)\begin{bmatrix}x^{\top}&1\end{bmatrix}^{\top},$

then the real variety
$\mathcal{V}_{\mathbb{R}}(\tilde{f}_{1}^{h},f_{2}^{h},f_{3}^{h})$ is still
empty. However, no nonzero aggregation lies in $\mathcal{P}$ and the convex
hull of the set defined by the inequalities is given by aggregations. This is
shown in Figure 2.

Figure 2. The spectral curve and its relation to the modified system of
inequalities (2) (left). Note that the conical hull of the defining
inequalities has empty intersection with the hyperbolicity cone of $g$ and
that the intersection of $S_{\lambda}$ for $\lambda\in\Lambda^{\prime}$ gives
$\mathop{\overline{\rm conv}}(S)$ (right).

Finally, we consider the case where $n=2$,
$\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})=\emptyset$, but $g$
is not hyperbolic. In this setting, we have that for any plane
$H\subseteq\mathbb{R}^{3}$, the restrictions
$Q_{1}|_{H},Q_{2}|_{H},Q_{3}|_{H}$ satisfy PDLC. It follows that
$\mathop{\overline{\rm conv}}(S)$ is given by good aggregations, but it may be
necesssary to use infinitely many.

######  Theorem 3.9.

Suppose that $n=2$,
$\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})=\emptyset$, and $g$
is smooth but not hyperbolic. Then there is a (possibly infinite) set
$\Lambda_{1}\subseteq\mathbb{R}^{3}_{+}$ of good aggregations such that
$\mathop{\overline{\rm conv}}(S)=\bigcap_{\lambda\in\Lambda_{1}}S_{\lambda}$.

Theorem 3.9 is demonstrated in the following example from [6].

######  Example 3.10 ([6, Proposition 2.8]).

Consider the matrices

$Q_{1}=\begin{bmatrix}1&0&0\\\ 0&0&0\\\ 0&0&-1\end{bmatrix},\quad
Q_{2}=\begin{bmatrix}0&0&0\\\ 0&1&0\\\
0&0&-1\end{bmatrix},\quad\text{and}\quad Q_{3}=\begin{bmatrix}-1&0&1\\\
0&-1&1\\\ 1&1&-1\end{bmatrix},$

so that

$S=\\{(x,y)\in\mathbb{R}^{2}\;|\;x^{2}-1\leq 0,y^{2}-1\leq
0,-(x-1)^{2}-(y-1)^{2}+1\leq 0\\}.$

The determinant is

$g(\lambda)=\det\begin{bmatrix}\lambda_{1}-\lambda_{3}&0&\lambda_{3}\\\
0&\lambda_{2}-\lambda_{3}&\lambda_{3}\\\
\lambda_{3}&\lambda_{3}&-(\lambda_{1}+\lambda_{2}+\lambda_{3})\end{bmatrix}=-\lambda_{1}^{2}\lambda_{2}-\lambda_{1}\lambda_{2}^{2}+\lambda_{1}^{2}\lambda_{3}+\lambda_{1}\lambda_{2}\lambda_{3}+\lambda_{2}^{2}\lambda_{3}-\lambda_{1}\lambda_{3}^{2}-\lambda_{2}\lambda_{3}^{2}+\lambda_{3}^{3}$

The affine cone over the spectral curve is displayed below in Figure 3. It is
smooth, but $g$ is not hyperbolic. As shown in [6, Proposition 2.8], we have
that $\mathop{\overline{\rm conv}}(S)$ is given by the intersection of
infinitely many good aggregations and is a proper subset of the intersection
of any finite number of good aggregations.

Figure 3. The affine cone over the spectral curve in Example 3.10. The
spectral curve is smooth but not hyperbolic. In this case,
$\mathop{\overline{\rm conv}}(S)$ is obtained as the intersection of good
aggregations, but any finite subset of good aggregations will give a strict
superset of $\mathop{\overline{\rm conv}}(S)$.

###  3.3. Bounds on the Number of Needed Aggregations

Using the tools developed in Section 3.1, we show that if the real projective
variety $\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})$ is empty and
the spectral curve is hyperbolic, then a finite number of permissible
aggregations suffices to recover the set defined by all permissible
aggregations. An immediate reduction is that it suffices to consider only unit
length generators of the extreme rays of the set $\Lambda$. To see this, note
that $S_{\lambda}=S_{t\lambda}$ for $t>0$ and if $\omega\in\Lambda$ and
$\omega=t\lambda^{(1)}+(1-t)\lambda^{(2)}$ for some $t\in(0,1)$ and
$\lambda^{(1)},\lambda^{(2)}\in\Lambda$, then $S_{\omega}\subseteq
S_{\lambda^{(1)}}\cap S_{\lambda^{(2)}}$ since

$f_{\omega}(x)=f_{t\lambda^{(1)}+(1-t)\lambda^{(2)}}(x)=tf_{\lambda^{(1)}}(x)+(1-t)f_{\lambda^{(2)}}(x)\leq
0.$

We further reduce the number of needed aggregations by applying the following
property.

######  Lemma 3.11.

Let $\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(k+1)}\in\Lambda$. Set
$f_{k}^{h}=(f_{\lambda^{(1)}}^{h},\ldots,f_{\lambda^{(k)}}^{h})$ and
$f_{k+1}^{h}=(f_{\lambda^{(1)}},\ldots,f_{\lambda^{(k+1)}}^{h})$. Suppose that
$\mathcal{V}_{\mathbb{R}}(f_{\lambda^{(k+1)}}^{h})\cap
X(-\mathbb{R}^{k+1}_{+},f_{k+1}^{h})=\emptyset$ and that
$X(-\mathbb{R}^{k+1}_{+},f_{k+1}^{h})$ has the same number of connected
components as $X(-\mathbb{R}^{k}_{+},f_{k}^{h})$. Then,

$X(-\mathbb{R}^{k+1}_{+},f_{k+1}^{h})=X(-\mathbb{R}^{k}_{+},f_{k}^{h}).$

Lemma 3.11 allows us to conclude that an aggregation is not necessary by
applying Proposition 3.4 with a cone of the form
$K=\mathbb{R}^{k}\times\\{0\\}$. By the hypothesis that the determinantal
polynomial $g$ is hyperbolic, the computation of $H^{1}(\Omega^{n}(K))$
reduces to arguments about separating a polyhedral cone from a hyperbolicity
cone of $g$. Let $\mathrm{ex}(\Lambda)$ be the set of generators of extreme
rays of $\Lambda$ with unit length and set

$\Lambda^{\prime}=\\{\lambda\in\mathrm{ex}(\Lambda)\;|\;|\operatorname*{supp}(\lambda)|\leq
2\\}.$

Note that $|\Lambda^{\prime}|\leq 6$ since there can be at most two extreme
rays of $\Lambda$ contained in each facet of $\mathbb{R}^{3}_{+}$. Under a
condition that ensures connectedness, $\Lambda^{\prime}$ suffices to describe
all permissible aggregations. Specifically, we enforce that
$\Omega^{n}(\mathop{\rm cone}(\Lambda^{\prime})^{\circ})$ is contractible,
i.e, the set of matrices $Q_{\lambda}$ for $\lambda\in\mathop{\rm
cone}(\Lambda^{\prime})\cap\mathbb{S}^{2}$ which have at least $n$ positive
eigenvalues is connected and has no higher cohomology. Using the spectral
sequence of Theorem 3.1, we see that the condition that
$\Omega^{n}(\mathop{\rm cone}(\Lambda^{\prime})^{\circ})$ is contractible is
met when $\bigcap_{\lambda\in\Lambda^{\prime}}S_{\lambda}$ is connected. In
particular, $\Omega^{n}(\mathop{\rm cone}(\Lambda^{\prime})^{\circ})$ is
contractible when $\bigcap_{\lambda\in\Lambda^{\prime}}S_{\lambda}$ is convex.

######  Theorem 3.12.

Suppose that
$\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})\subseteq\mathbb{RP}^{n}$
is empty, the spectral curve is smooth and hyperbolic, and that
$\Omega^{n}(\mathop{\rm cone}(\Lambda^{\prime})^{\circ})$ is contractible.
Then,

$\bigcap_{\lambda\in\Lambda^{\prime}}S_{\lambda}=\bigcap_{\lambda\in\Lambda}S_{\lambda}.$

We show in Example 3.13 that if the contractibility condition on
$\Omega^{n}(\mathop{\rm cone}(\Lambda^{\prime})^{\circ})$ is not satisfied,
then it may be necessary to use an aggregation with strictly positive entries.
In this case, the aggregation with strictly positive entries has the effect of
cutting off a connected component of
$\bigcap_{\lambda\in\Lambda^{\prime}}S_{\lambda}$.

######  Example 3.13.

We construct a system of quadratics such that
$\bigcap_{\lambda\in\Lambda^{\prime}}S_{\lambda}$ has three connected
components, but for some $\mu$ with strictly positive entries,
$S_{\mu}\cap\left(\bigcap_{\lambda\in\Lambda^{\prime}}S_{\lambda}\right)$ has
only two components. As in Example 3.2, consider the matrices

$M_{1}=\begin{bmatrix}25&0&-32&0\\\ 0&25&0&24\\\ -32&0&6&0\\\
0&24&0&6\end{bmatrix},\quad M_{2}=\begin{bmatrix}0&0&12&0\\\ 0&0&0&16\\\
12&0&4&0\\\
0&16&0&4\end{bmatrix},\;\text{and}\;M_{3}=\begin{bmatrix}0&0&0&-60\\\
0&0&10&0\\\ 0&10&0&0\\\ -60&0&0&0\end{bmatrix}.$

The matrices do not satisfy PDLC and $\det(xM_{1}+yM_{2}+zM_{3})$ is
hyperbolic with respect to $(1,0,0)$. We construct the defining quadratics
$f_{i}(x,y,z)=\begin{bmatrix}x&y&z&1\end{bmatrix}Q_{i}\begin{bmatrix}x&y&z&1\end{bmatrix}^{\top}$
with $Q_{i}$ defined as follows:

$\displaystyle Q_{1}=M_{1}+1.5M_{2}=\begin{bmatrix}25&0&-14&0\\\ 0&25&0&48\\\
-14&0&12&0\\\ 0&48&0&12\end{bmatrix}$ $\displaystyle
Q_{2}=M_{1}-2M_{2}+2M_{3}=\begin{bmatrix}25&0&-56&-120\\\ 0&25&20&-8\\\
-56&20&-2&0\\\ -120&-8&0&-2\end{bmatrix}$ $\displaystyle
Q_{3}=M_{1}-2M_{2}-2M_{3}=\begin{bmatrix}25&0&-56&120\\\ 0&25&-20&-8\\\
-56&-20&-2&0\\\ 120&-8&0&-2\end{bmatrix}$

We numerically compute that the elements of $\Lambda^{\prime}$ are
appropriately normalized vectors in the directions of the following vectors
$\lambda^{(1)},\lambda^{(2)},\lambda^{(3)}$. Since the semialgebraic set
$S_{\lambda}$ is invariant under positive scalings of $\lambda$, we work
directly with the $\lambda^{(i)}$. We also take an aggregation $\mu$ which
lies on the oval of depth $\lfloor\frac{n+1}{2}\rfloor-1$ with strictly
positive entries.

$\lambda^{(1)}=e_{1},\quad\lambda^{(2)}=(1,0.68725,0),\quad\lambda^{(3)}=(1,0,0.68725),\quad\mu=(0.1429,0.4286,0.4286)$

The intersection of the cone $K=\mathop{\rm
cone}(\lambda^{(1)},\lambda^{(2)},\lambda^{(3)})$ with
$\Omega^{n}(-\mathbb{R}^{3}_{+})$ has three connected components and the
intersection of the cone $\tilde{K}=\mathop{\rm
cone}(\lambda^{(1)},\lambda^{(2)},\lambda^{(3)},\mu)$ with
$\Omega^{n}(-\mathbb{R}^{3}_{+})$ has two connected components. The
corresponding semialgebraic sets $\bigcap_{i=1}^{3}S_{\lambda^{(i)}}$ and
$S_{\mu}\cap\left(\bigcap_{i=1}^{3}S_{\lambda^{(i)}}\right)$ have three and
two connected components, respectively. However, as in the proof of Theorem
3.12,
$\mathcal{V}_{\mathbb{R}}(f_{\mu}^{h})\cap\left(\bigcap_{i=1}^{3}S_{\lambda^{(i)}}\right)=\emptyset$.
So intersecting with $S_{\mu}$ has the effect of cutting off a component of
$\left(\bigcap_{i=1}^{3}S_{\lambda^{(i)}}\right)$. In particular, we see that
$\bigcap_{\lambda\in\Lambda^{\prime}}S_{\lambda}\not=\bigcap_{\lambda\in\Lambda}S_{\lambda}$.
The spectral curve and the relevant semialgebraic sets are shown in Figure 4.

Figure 4. Plots for Example 3.13. The left figure displays the spectral curve
and relevant aggregations and cones. The center figure shows the set
$S_{\mu}\cap\left(\bigcap_{\lambda\in\Lambda^{\prime}}S_{\lambda}\right)$
which has two connected components. The right figure shows
$\bigcap_{\lambda\in\Lambda^{\prime}}S_{\lambda},$ which has three connected
components.

The construction in Example 3.13 inspires the following bound on the number of
needed aggregations in terms of the number of connected components of the set
$\Omega^{n}(\mathop{\rm cone}(\Lambda^{\prime})^{\circ})$.

######  Theorem 3.14.

Suppose that
$\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})\subseteq\mathbb{RP}^{n}$
is empty, the spectral curve is smooth and hyperbolic, and
$\Omega^{n}(\mathop{\rm cone}(\Lambda^{\prime})^{\circ})$ is not contractible.
Then, there is a set $\Lambda_{1}\subseteq\Lambda$ with
$|\Lambda_{1}|\leq|\Lambda^{\prime}|+\dim(H^{0}(\Omega^{n}(\mathop{\rm
cone}(\Lambda^{\prime})^{\circ})))-1$ such that

$\bigcap_{\lambda\in\Lambda_{1}}S_{\lambda}=\bigcap_{\lambda\in\Lambda}S_{\lambda}.$

###  3.4. Topology of Good Aggregations

Our final goal is an investigation of the topology of good and bad
aggregations.

######  Proposition 3.15.

Suppose that $S=\mathrm{cl}(\mathrm{int}(S))$. Then, every connected component
of $\Lambda$ consists of only good aggregations or of only bad aggregations.

The assumption that $S$ satisfies the regularity condition
$S=\mathrm{cl}(\mathrm{int}(S))$ cannot be removed, as demonstrated by the
following example.

######  Example 3.16.

This example is modified from [3, Example 2.22] Consider the three matrices

$Q_{1}=\begin{bmatrix}-1&0&0.5\\\ 0&0&0\\\ 0.5&0&0\end{bmatrix},\qquad
Q_{2}=\begin{bmatrix}1&0&0\\\ 0&1&0\\\ 0&0&-1\end{bmatrix},\quad\text{ and
}\quad Q_{3}=\begin{bmatrix}1&0&0\\\ 0&1&0\\\ 0&0&-3\end{bmatrix}$

and the associated quadratic functions $f_{i}$. Note that the linear
combination

$Q_{1}+4Q_{2}-2Q3=\begin{bmatrix}1&0&0.5\\\ 0&2&0\\\ 0.5&0&2\end{bmatrix}$

is positive definite. The set

$S=\left\\{x=(x_{1},x_{2})\in\mathbb{R}^{2}\;|\;\|x\|_{2}^{2}\leq 1,x_{1}\leq
0\right\\}\cup\\{e_{1}\\}$

does not satisfy $S=\mathrm{cl}(\mathrm{int}(S))$. Now, $\Lambda$ is
connected, but there exist both good and bad aggregations ($(0.5,0.5,0)$ is a
good aggregation while $(0.75,0.25,0)$ is a bad aggregation). On the other
hand, $\Omega^{n}(-\mathbb{R}^{3}_{+})$ has two connected components.
Additionally, note that every element $\lambda\in\Lambda$ satisfies the
property $\mathop{\rm
conv}(\mathrm{int}(S))\subseteq\mathrm{int}(S_{\lambda})$.

Finally, under the assumption of PDLC, there is a unique connected component
of $\Omega^{n}(-\mathbb{R}^{3}_{+})$ consisting of good aggregations.

######  Proposition 3.17.

If $Q_{1},Q_{2},Q_{3}$ satisfy PDLC, $\mathrm{int}(S)\not=\emptyset$, and $S$
has no points at infinity, then exactly one connected component of
$\Omega^{n}(-\mathbb{R}^{3}_{+})$ consists of good aggregations.

Using the topology of good aggregations as a subset of all permissible
aggregations, we prove Theorem 1.4. Specifically, under PDLC and regularity
hypotheses on $S$, there is a set of $r\leq 4$ good aggregations
$\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(r)}$ such that
$\bigcap_{i=1}^{r}S_{\lambda^{(i)}}$. The proof is in Section 8.

##  4\. Conclusions

We have presented an analysis of aggregations of three quadratic inequalities
under the assumption that the projective variety of the defining quadratics is
empty. In particular, we provide a sufficient condition for the description of
$\mathop{\overline{\rm conv}}(S)$ via good aggregations which does not depend
on PDLC. We show that the semialgebraic set defined by the intersection of all
permissible aggregations can be described by the intersection of finitely many
permissible aggregations when the spectral curve is hyperbolic. If the set is
connected, then we can explicitly compute these aggregations.

Our approach differs from previous approaches to this problem in that we
heavily utilize tools from algebraic topology. These tools are infrequent in
the optimziation literature, but this work demonstrates that these topological
tools can have useful applications. Our analysis is limited to the case of
three inequalities and an empty variety, as we heavily rely on the structure
of hyperbolic plane curves. We leave open the case of more defining
inequalities or a nonempty variety. In these cases, the spectral hypersurface
may be non-hyperbolic or not a plane curve, preventing a direct translation of
our results.

##  5\. Spectral Sequence Computations

In this section, we present the spectral sequence computations used to derive
the results in Section 3.1. We first concentrate on the case that $n\geq 4$,
$K=\\{0\\}$, and
$X=\mathcal{V}(f_{1}^{h},f_{2}^{h},f_{3}^{h})\subseteq\mathbb{RP}^{n}$ and
prove that the emptiness of $X$ implies hyperbolicity of
$g(\lambda)=\det(Q_{\lambda})$. Note that the case $n=3$ is proved in [13,
Theorem 7.8].

We begin by relating the signature of a matrix $Q_{\lambda}$ to the location
of $\lambda$ relative to the ovals of the spectral curve
$\mathcal{V}_{\mathbb{R}}(g)$ when it is smooth and nonempty.

######  Lemma 5.1.

If $\lambda\in\Omega^{n}(\\{0\\})$ and the spectral curve
$\mathcal{V}_{\mathbb{R}}(g)$ is smooth, then $[\lambda]\in\mathbb{RP}^{2}$
lies in the interior of an oval of $\mathcal{V}_{\mathbb{R}}(g)$ of depth at
least $\lfloor\frac{n+1}{2}\rfloor-1$.

###### Proof.

Let $x\in\mathbb{S}^{2}$ be such that $[x]\in\mathbb{RP}^{2}$ lies on the
exterior of every oval of $\mathcal{V}_{\mathbb{R}}(g)$. By [16], if $n+1$ is
even $Q_{x}$ has $\frac{n+1}{2}$ positive and negative eigenvalues and if
$n+1$ is odd, one of $Q_{x}$ and $-Q_{x}$ has $\lfloor\frac{n+1}{2}\rfloor+1$
positive and $\lfloor\frac{n+1}{2}\rfloor$ negative eigenvalues. Since $g$ is
nonsingular, it follows that if $[\lambda]$ is in the interior of an oval of
depth $k$ and the exterior of all ovals of depth $k+1$, then $Q_{\lambda}$ has
at most $\frac{n+1}{2}+k$ positive eigenvalues if $n+1$ is even and
$\lfloor\frac{n+1}{2}\rfloor+k+1$ positive eigenvalues if $n+1$ is odd. So, if
$Q_{\lambda}$ has $n$ positive eigenvalues and $n+1$ is even, $[\lambda]$ lies
in the interior of an oval of depth at least $\frac{n+1}{2}-1$ and if $n+1$ is
odd, $[\lambda]$ is an element of the interior of an oval of depth at least
$\lfloor\frac{n+1}{2}\rfloor-1$ ∎

The spectral sequence of Theorem 3.1 then gives the following.

######  Lemma 5.2 (cf [2, Corollary 2]).

Suppose that $n\geq 4$ and the spectral curve is smooth. If
$X=\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})\subseteq\mathbb{RP}^{n}$
is empty, then $g$ is hyperbolic and $\Omega^{n}(\\{0\\})\not=\emptyset$.

###### Proof.

For notational simplicity, take $X=X(\\{0\\},f^{h})$ and
$\Omega^{j}=\Omega^{j}(\\{0\\})$. Let $(E_{r},d_{r})$ be the spectral sequence
of Theorem 3.1.

First, suppose that the matrices $Q_{i}$ satisfy PDLC so that there is
$\lambda\in\mathbb{R}^{3}$ with $Q_{\lambda}\succ 0$. Then $g$ is hyperbolic
with respect to $\lambda$ and $Q_{\lambda}$ has $n+1>n$ positive eigenvalues
so that $\Omega^{n}\not=\emptyset$.

Next we show that if the maximum number of positive eigenvalues attained by
$Q_{\lambda}$ for $\lambda\in\mathbb{R}^{3}$ is at most $n-1$, then
$X\not=\emptyset$. Using (1), we see that $E^{ij}_{2}=H^{i-1}(\Omega^{j+1})=0$
for $i\geq 4$ and that $E_{2}^{2,n-1}=0$. Note that $\Omega^{n-1}$ is either
empty or a proper subset of $\mathbb{S}^{2}$. This follows since $n\geq 4$ and
therefore if $\lambda$ is such that $Q_{\lambda}$ has at least $n-1$ positive
eigenvalues, then $Q_{-\lambda}$ has at most $(n+1)-(n-1)=2<n-1$ positive
eigenvalues. So $E_{2}^{3,n-2}=H^{2}(\Omega^{n-1})=0$. This implies that
$E_{2}^{i,n-i+1}=0$ for all $i\geq 2$ and therefore $E_{2}^{0,n}\simeq
E_{\infty}^{0,n}\simeq\mathbb{Z}_{2}$. So, $H_{0}(X)\not=0$ and therefore $X$
is nonempty.

Finally, suppose that there exists $\lambda\in\mathbb{R}^{3}$ such that
$Q_{\lambda}$ has exactly $n$ positive eigenvalues and that PDLC is not
satisfied. We apply Theorem 3.1 and the hypothesis that $X=\emptyset$ to
compute the groups $H^{i}(\Omega^{j})$. Note that $Q_{-\lambda}$ has at most 1
positive eigenvalue so that $\Omega^{j+1}$ is a proper subset of
$\mathbb{S}^{2}$ for $j\geq 1$. Since $n\geq 4$, this implies that
$E^{3,j}_{2}\simeq H^{2}(\Omega^{j+1})=0$ for $j\geq n-3$. So, the $E_{2}$
page of the spectral sequence has the following form:

[title = $E_{\page}$, no y ticks, name = empty var top four, cohomological
Serre grading, classes = draw = none , xscale = 2] [background] at (-0.6,0)
n-3; at (-0.6,1) n-2; at (-0.6,2) n-1; at (-0.6,3) n; [”Z_2”](0,3) [”0”](0,0)
[”0”](0,1) [”0”](0,2)

[”0”](1,3) [”H^0(Ω^n)/Z_2”](1,2) [”H^0(Ω^n-1)/Z_2”](1,1)
[”H^0(Ω^n-2)/Z_2”](1,0)

[”0”](2,3) [”H^1(Ω^n)”](2,2) [”H^1(Ω^n-1)”](2,1) [”H^1(Ω^n-2)”](2,0)

[”0”](3,0) [”0”](3,1) [”0”](3,2) [”0”](3,3)

$\printpage[name=emptyvartopfour,page=2]$

Note that $E_{2}^{1,n-1}\simeq E_{\infty}^{1,n-1}\simeq
H^{0}(\Omega^{n})/\mathbb{Z}_{2}$ and $E_{2}^{2,n-2}\simeq
E_{\infty}^{2,n-2}\simeq H^{1}(\Omega^{n-1})$. Since $X=\emptyset$, Theorem
3.1 implies that

$0=H_{0}(X)\simeq\bigoplus_{i+j=n}E^{i,j}_{\infty}.$

In particular, $H^{0}(\Omega^{n})/\mathbb{Z}_{2}=0$, $H^{1}(\Omega^{n-1})=0$,
and the map $d_{2}^{0,n}:\mathbb{Z}_{2}\to H^{1}(\Omega^{n})$ is injective. By
Lemma 5.1, $\Omega^{n}$ is a subset of the interior of ovals of depth at least
$\lfloor\frac{n+1}{2}\rfloor-1$. Since $H^{1}(\Omega^{n})\not=0$, there is
anest ovals of depth $\lfloor\frac{n+1}{2}\rfloor$. So, the curve
$\mathcal{V}_{\mathbb{R}}(g)$ is hyperbolic. ∎

For the converse statement, note that if PDLC holds, then the variety is
empty. The remaining case is that $g$ is hyperbolic with a hyperbolicity cone
$\mathcal{P}$ which consists of matrices with exactly $n-1$ positive and two
negative eigenvalues. In this case, we must compute the differential
$d_{2}^{0,n}$. In particular, we want to show that
$d_{2}^{0,n}:\mathbb{Z}_{2}\to H^{1}(\Omega^{n})$ is injective. Let
$\mathcal{Q}$ be the vector space of quadratic forms on $\mathbb{R}^{n+1}$.
For $q\in\mathcal{Q}$, let
$\rho_{1}(q)\geq\rho_{2}(q)\geq\ldots\geq\rho_{n+1}(q)$ be the eigenvalues of
$q$ and $\mathcal{D}_{j}=\\{q\in\mathcal{Q}\;|\;\rho_{j}(q)>\rho_{j+1}(q)\\}.$

######  Theorem 5.3 ([1, Theorem B]).

There is a formula for the differential $d_{2}$ in terms of matrices with
repeated eigenvalues. Explicitly,

$d_{2}(x)=(x\smile\bar{f}^{*}\gamma_{1,j})|_{(C\Omega,\Omega^{j})}\quad\text{
for }x\in H^{*}(C\Omega,\Omega^{j+1}),$

where $\bar{f}:\mathbb{R}^{3}\to\mathcal{Q}$ is defined by
$\bar{f}(\lambda)=f_{\lambda}^{h}$, and $\bar{f}^{*}$ is the induced map on
cohomology. The value of the class $\gamma_{1,j}\in
H^{2}(\mathcal{Q},\mathcal{D}_{j})$ on the image of
$\sigma:B^{2}\to\mathcal{Q}$ with $\sigma(\partial
B^{2})\subseteq\mathcal{D}_{j}$ is equal to the intersection number of
$\sigma(B^{2})$ and $\mathcal{Q}\setminus\mathcal{D}_{j}\mod 2$.

In light of Theorem 5.3, the differential $d_{2}^{0,n}$ is computed by
understanding the set of matrices with repeated negative eigenvalue inside the
hyperbolicity cone $\mathcal{P}$ of $g$. To show that $d_{2}^{0,n}$ is
injective, it suffices to show that there is exactly one
$\lambda\in\mathrm{int}\mathcal{P}\cap\mathbb{S}^{2}$ with repeated negative
eigenvalue. In this case, it will follow that $\gamma_{1,n}$ and therefore
$d_{2}^{0,n}(1)$ are nonzero. We proceed in several steps: Lemma 5.4 and
Proposition 5.5 give the existence and uniqueness of
$\lambda\in\mathrm{int}\mathcal{P}\cap\mathbb{S}^{2}$ with repeated negative
eigenvalue, which is leveraged for the computation of $d_{2}^{0,n}(1)$ in the
proof of Theorem 3.3.

We first show that if the matrices corresponding to two points in
$\mathcal{P}$ have a repeated negative eigenvalue, then the matrices
corresponding to every point along the affine pencil connecting them has at
least 2 negative eigenvalues.

######  Lemma 5.4.

Suppose that $\Omega^{n}\not=\emptyset$ and $g$ is smooth and hyperbolic with
hyperbolicity cone $\mathcal{P}$ containing matrices with exactly two negative
eigenvalues. If $\lambda^{(1)},\lambda^{(2)}\in\mathcal{P}$ are such that
$Q_{\lambda^{(1)}}$ and $Q_{\lambda^{(2)}}$ have repeated negative eigenvalue
$-1$, then for each $t\in\mathbb{R}$,
$tQ_{\lambda^{(1)}}+(1-t)Q_{\lambda^{(2)}}$ has at least 2 negative
eigenvalues.

###### Proof.

First, note that the statement holds for all $t\in[0,1]$ by convexity of
$\mathcal{P}$. Let $V$ be the two dimensional subspace corresponding to the
eigenvalue $-1$ of $Q_{\lambda^{(1)}}$ and $U$ be the two dimensional subspace
corresponding to the eigenvalue $-1$ of $Q_{\lambda^{(2)}}$. Then, for $t\geq
1$, and for all $v\in V$ with $v^{\top}v=1$, we have that

$v^{\top}(tQ_{\lambda^{(1)}}+(1-t)Q_{\lambda^{(2)}})v=-t+(1-t)v^{\top}Q_{\lambda^{(2)}}v\leq-1,$

where the inequality holds since $-1$ is the lowest eigenvalue of
$Q_{\lambda^{(2)}}$ and therefore $v^{\top}Q_{\lambda^{(2)}}v\geq-1$.
Similarly, if $u\in U$ has $u^{\top}u=1$ and $t\leq 0$, then

$u^{\top}(tQ_{\lambda^{(1)}}+(1-t)Q_{\lambda^{(2)}})u=tu^{\top}Q_{\lambda^{(1)}}u-(1-t)=t(u^{\top}Q_{\lambda^{(1)}}u+1)-1\leq-1,$

where the inequality holds since $u^{\top}Q_{\lambda^{(1)}}u\geq-1$.

The claim then follows from the variational characterization of eigenvalues. ∎

We now show that if the hyperbolicity cone $\mathcal{P}$ is such that
$\lambda\in\mathrm{int}\mathcal{P}$ implies that $Q_{\lambda}$ has exactly two
negative eigenvalues, then there is a unique
$\lambda\in\mathcal{P}\cap\mathbb{S}^{2}$ such that $Q_{\lambda}$ has a
repeated negative eigenvalue. In particular, this will imply that the
differential $d_{2}^{0,n}$ is nontrivial.

######  Proposition 5.5.

Suppose that $\Omega^{n}\not=\emptyset$ and $g$ is smooth and hyperbolic with
hyperbolicity cone $\mathcal{P}$ containing matrices with exactly 2 negative
eigenvalues. Then, there is a unique $\lambda\in\mathcal{P}\cap\mathbb{S}^{2}$
such that $Q_{\lambda}$ has a repeated negative eigenvalue.

###### Proof.

First, there can be at most one such $\lambda$. Suppose for the sake of a
contradiction that there were $\lambda^{(1)},\lambda^{(2)}\in\mathcal{P}$ such
that $Q_{\lambda^{(1)}}$ and $Q_{\lambda^{(2)}}$ have repeated negative
eigenvalues and $\lambda^{(1)}$ is not a scalar multiple of $\lambda^{(2)}$.
By rescaling the $\lambda^{(i)}$ if necessary, we can take the repeated
negative eigenvalue to be $-1$. Lemma 5.4 then implies that for all
$t\in\mathbb{R}$, the matrix $tQ_{\lambda^{(1)}}+(1-t)Q_{\lambda^{(2)}}$ has
at least two negative eigenvalues. This implies that
$t\lambda^{(1)}+(1-t)\lambda^{(2)}\in\mathrm{int}(\mathcal{P})$ for all
$t\in\mathbb{R}$ since $\omega\in\partial\mathcal{P}$ impies that $Q_{\omega}$
has $n-1$ positive eigenvalues, one negative eigenvalue, and $0$ as an
eigenvalue of multiplicity 1. In particular,
$g(t\lambda^{(1)}+(1-t)\lambda^{(2)})\not=0$ for $t\in\mathbb{R}$. On the
other hand, $g$ is hyperbolic with respect to $\lambda^{(2)}$, so that
$g(\lambda^{(2)}+t(\lambda^{(1)}-\lambda^{(2)}))\in\mathbb{R}[t]$ is real
rooted. So, the homogenized polynomial
$g(s\lambda^{(2)}+t(\lambda^{(1)}-\lambda^{(2)}))\in\mathbb{R}[s,t]$ must have
a zero of multiplicity $n+1$ at the point $[0:1]\in\mathbb{RP}^{1}$, a
contradiction with the hypothesis that $g$ is smooth. So, there is at most one
$\hat{\lambda}\in\mathcal{P}$ such that $Q_{\hat{\lambda}}$ has $-1$ as a
repeated negative eigenvalue and therefore there is at most one
$\lambda=\hat{\lambda}/\|\hat{\lambda}\|\in\mathcal{P}\cap\mathbb{S}^{2}$ such
that $Q_{\lambda}$ has a repeated negative eigenvalue.

We now show that there is exactly one such $\lambda$. Our strategy is to
choose a coordinate system where a matrix with repeated negative eigenvalue
exists. Let $\lambda\in\mathrm{int}\mathcal{P}\cap\mathbb{S}^{2}$ and let
$v^{(1)},v^{(2)},\ldots,v^{(n+1)}$ be an orthonormal basis of eigenvectors of
$Q_{\lambda}$ corresponding to the eigenvalues
$\rho_{1}(Q_{\lambda})\geq\rho_{2}(Q_{\lambda})\geq\ldots\geq\rho_{n+1}(Q_{\lambda})$.
Set
$B=\begin{bmatrix}\frac{1}{\sqrt{|\rho_{1}(Q_{\lambda})|}}v^{(1)}&\frac{1}{\sqrt{|\rho_{1}(Q_{\lambda})|}}v^{(2)}&\dots&\frac{1}{\sqrt{|\rho_{n+1}(Q_{\lambda})|}}v^{(n+1)}\end{bmatrix}$.
Then, $x\mapsto Bx$ defines a change of coordinates on $\mathbb{RP}^{n}$. In
these coordinates, the quadratic form $f_{\lambda}^{h}$ is represented by the
diagonal matrix

$B^{\top}Q_{\lambda}B=\begin{bmatrix}1&&&&\\\ &\ddots&&&\\\ &&1&&\\\ &&&-1&\\\
&&&&-1\end{bmatrix}$

Next, note that for $\omega\in\mathbb{R}^{3}$,
$\det(B^{\top}Q_{\omega}B)=\det(B)^{2}g(\omega)$ so that the zero set and
hyperbolicity cone of $g$ are preserved. By the above discussion, $\lambda$ is
the unique element of $\mathcal{P}\cap\mathbb{S}^{2}$ such that
$B^{\top}Q_{\lambda}B$ has a repeated negative eigenvalue. Theorem 5.3 then
implies that the variety
$\mathcal{V}_{\mathbb{R}}(B^{\top}Q_{1}B,B^{\top}Q_{2}B,B^{\top}Q_{3}B)\subseteq\mathbb{RP}^{n}$
is empty. Since nonexistence of real points on a variety is preserved under a
real coordinate change of $\mathbb{RP}^{n}$, this implies that
$\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})$ is also be empty. By
Theorem 5.3 and the preceding discussion, this implies that there is an odd
number of and therefore exactly one $\lambda\in\mathcal{P}\cap\mathbb{S}^{2}$
such that $Q_{\lambda}$ has a repeated negative eigenvalue. ∎

We treat the small $n$ cases separately and prove the theorem.

######  Lemma 5.6.

Suppose that $n=1$ and that $g(\lambda)$ is smooth. Then,
$\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})=\emptyset$ if and
only if PDLC holds.

###### Proof.

If PDLC holds, then
$X=\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})=\emptyset$. Suppose
that PDLC does not hold. Then, by the hypothesis that $g$ is smooth, we have
that $Q_{\lambda}$ has one positive and one negative eigenvalue for all
$\lambda\in\mathbb{R}^{3}$ and therefore
$H^{2}(\Omega^{1})\cong\mathbb{Z}_{2}$. So, if $(E_{r},d_{r})$ is the spectral
sequence of Theorem 3.1, then the $E_{2}$ page has the form

[title = $E_{\page}$, name = empty var n = 1, cohomological Serre grading,
classes = draw = none ]

[”Z_2”](0,1) [”0”](0,0)

[”0”](1,1) [”0”](1,0)

[”0”](2,1) [”0”](2,0)

[”0”](3,1) [”Z_2”](3,0)

$\printpage[name=emptyvarn=1,page=2]$

so that $0=d_{2}^{0,1}=d_{r}^{0,1}$ for all $r$. In particular, this implies
that $E_{\infty}^{0,1}\simeq\mathbb{Z}_{2}$ and therefore
$H_{0}(X)=\bigoplus_{i+j=1}E_{\infty}^{i,j}\not=0$ and $X\not=\emptyset$. ∎

There is also an elementary proof of Lemma 5.6 which follows by observing that
when $n=1$, the space of quadratics has dimension 3, so if the quadratic forms
$f_{1}^{h},f_{2}^{h},f_{3}^{h}$ are linearly independent, they span the entire
space of quadratics and therefore satisfy PDLC.

######  Lemma 5.7.

Suppose that $n=2$. If
$\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})\not=\emptyset$, then
$g(\lambda)$ is not smooth.

###### Proof.

Suppose that $v\in\mathbb{R}^{3}$ is nonzero and $f_{i}^{h}(v)=0$ for
$i=1,2,3$. Then, we have that the vectors $Q_{1}v,Q_{2}v,Q_{3}v$ are all
orthogonal to $v$ and therefore $\dim(\mathop{\rm
span}\nolimits_{\mathbb{R}}\\{Q_{1}v,Q_{2}v,Q_{3}v\\})\leq 2$. So, there is
$\alpha\in\mathbb{R}^{3}$ such that
$0=\sum_{i=1}^{3}\alpha_{i}Q_{i}v=Q_{\alpha}v$. In particular, $Q_{\alpha}$
has rank at most two. If $Q_{\alpha}$ is rank one, then $\alpha$ is a singular
point of $\mathcal{V}_{\mathbb{R}}(g)$. Suppose that $Q_{\alpha}$ has rank
two. Let $\mathcal{D}\subseteq\mathrm{Sym}_{3}(\mathbb{R})$ be the
hypersurface in the space of $3\times 3$ symmetric real matrices defined by
the vanishing of the determinant. Note that $Q_{\alpha}\in\mathcal{D}$. By
Jacobi’s formula, we have that
$\nabla\det(Q_{\alpha})=\mathrm{adj}Q_{\alpha}=cvv^{\top}$ for some constant
$c$. This implies that $T_{Q_{\alpha}}\mathcal{D}$, the tangent space to
$\mathcal{D}$ at $Q_{\alpha}$, is the space of quadratic forms which vanish at
$v$, as $\langle\mathrm{adj}Q_{\alpha},Q\rangle=0$ implies that
$\mathrm{tr}(vv^{\top}Q)=v^{\top}Qv=0$. But then, $Q_{i}\in
T_{Q_{\alpha}}\mathcal{D}$ for each $i\in[3]$. So, $\mathop{\rm
span}\nolimits_{\mathbb{R}}(\\{Q_{i}\;|\;i\in[3]\\})$ intersects $\mathcal{D}$
non-transversely at $Q_{\alpha}$ and therefore $\alpha$ is a singular point of
$g$. ∎

We are now ready to prove Theorem 3.3.

See 3.3

###### Proof.

The cases $n=1$, $n=2$, and $n=3$ are treated by Lemma 5.6, Lemma 5.7, and
[13, Theorem 7.8], respectively. The “only if” statement for $n\geq 4$ is
Lemma 5.2 [2, Corollary 2]. If PDLC holds, then $X(\\{0\\},f^{h})=\emptyset$
for any $n$.

In the remaining case where PDLC does not hold and $n\geq 4$, we have that
$\Omega^{n}\not=\emptyset$ and $H^{1}(\Omega^{n})\simeq
H^{2}(C\Omega,\Omega^{n})\not=0$. Let $\sigma:B^{2}\to C\Omega$ be a
representative of the nontrivial class in $H^{2}(C\Omega,\Omega^{n})$. Then,
with notation as in the statement of Theorem 5.3,
$\gamma_{1,n}(\bar{f}(\sigma))$ is equal to the number of
$\lambda\in\mathcal{P}\cap\mathbb{S}^{2}$ such that $Q_{\lambda}$ has a
repeated negative eigenvalue. So, by Theorem 5.3 and Proposition 5.5, there is
exactly one such $\lambda$ and we compute that

$d_{2}^{0,n}(1)(\sigma)=1\smile\bar{f}^{*}\gamma_{1,n}(\sigma)=\gamma_{1,n}(\bar{f}(\sigma))=1.$

So, $d_{2}^{0,n}:\mathbb{Z}_{2}\to H^{1}(\Omega^{n})$ is injective and
therefore $E_{3}^{0,n}\simeq E_{\infty}^{0,n}\simeq 0$. This then implies that
$H_{0}(X)\simeq\bigoplus_{i+j=n}E_{\infty}^{0,n}\simeq 0$.∎

Together, the results of this section imply Theorem 1.1.

See 1.1

###### Proof.

The statement for $n=3$ is [13, Theorem 7.8]. The remaining statements are
Lemmas 5.2, 5.6, and 5.7 and Theorem 3.3. ∎

Theorem 3.3 places strong structure restrictions on the spectral curve
$\mathcal{V}_{\mathbb{R}}(g)$ in the case that $n\geq 3$ and the real variety
is empty. Certificates for emptiness of the projective variety
$\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})$ can in turn be
leveraged to construct certificates of emptiness for systems of inequalities,
as demonstrated in the following proposition.

See 3.4

###### Proof.

To simplify notation, we take $\Omega^{j}=\Omega^{j}(K)$ for all $j$ and
$X=X(K,f^{h})$. If $\Omega^{n+1}\neq\emptyset$, then there is $\lambda\in
K^{\circ}\cap S^{2}$ with $Q_{\lambda}\succ 0$ and therefore $X=\emptyset$.

Next, note that $\Omega^{n}\neq\emptyset$ is a necessary condition for
$X=\emptyset$. Otherwise, as in the proof of Lemma 5.2, the spectral sequence
of Theorem 3.1 would have that $E_{2}^{0,n}\simeq
E_{\infty}^{0,n}\simeq\mathbb{Z}_{2}$ is a direct summand of $H_{0}(X)$ and
therefore $H_{0}(X)\neq 0$.

Now, suppose that $\Omega^{n+1}=\emptyset$ and $\Omega^{n}\neq\emptyset$. Note
that the hypothesis that $K\not=0$ implies that $K^{\circ}\not=\mathbb{R}^{3}$
and therefore $H^{2}(\Omega^{j})=0$ for all $j$. So, the $E_{2}$ page of the
spectral sequence of Theorem 3.1 has first four columns

$\printpage[name=emptyvartopfour,page=2]$

So if $X=\emptyset$, then $d_{2}^{0,n}:\mathbb{Z}_{2}\to H^{1}(\Omega^{n})$ is
injective and therefore $H^{1}(\Omega^{n})\neq\emptyset$.

Conversely, suppose that $H^{1}(\Omega^{n})\neq\emptyset$. As in the proof of
Theorem 3.3, this implies that the polynomial $g$ is hyperbolic with a
hyperbolicity cone $\mathcal{P}$ such that $\lambda\in\mathcal{P}$ implies
that $Q_{\lambda}$ has exactly two negative eigenvalues. Moreover, there is
exactly one $\lambda\in\mathcal{P}\cap\mathbb{S}^{2}$ such that $Q_{\lambda}$
has a repeated negative eigenvalue. But then, the differential
$d_{2}^{0,n}:\mathbb{Z}^{2}\to H^{1}(\Omega^{n})$ is injective, which implies
that $E_{3}^{0,n}\simeq E_{\infty}^{0,n}\simeq 0$ and therefore $H_{0}(X)=0$.

∎

Proposition 3.4 allows us to certify emptiness of solutions sets of systems of
quadratic inequalities using convexity. In particular, if
$\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})=\emptyset$ with a
smooth hyperbolic spectral curve (as is guaranteed for $n\geq 3$), the
condition $\Omega^{n+1}(K)\not=\emptyset$ is equivalent to the statement that
there is a hyperbolicity cone $\mathcal{P}$ of $g$ which intersects
$K^{\circ}$ and that $\lambda\in\mathcal{P}$ implies $Q_{\lambda}\succ 0$.
Similarly, the condition $H^{1}(\Omega^{n}(K))\not=0$ is equivalent to the
conditions that $\mathcal{P}\setminus\\{0\\}\subseteq\mathrm{int}(K^{\circ})$
and $Q_{\lambda}$ has exactly two negative eigenvalues for
$\lambda\in\mathcal{P}$.

We conclude this section with a computation of the number of connected
components of the set $X(K,f^{h})$ which will be used in later sections.

######  Lemma 5.8.

Suppose that
$\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})=\emptyset$ and $K$ is
a nonzero polyhedral cone such that $X(K,f^{h})\not=\emptyset$ and that
$\Omega^{n}(K)\neq\emptyset$. Then, $H_{0}(X(K,f^{h}))\simeq
H^{0}(\Omega^{n}(K))$.

###### Proof.

Let $(E_{r},d_{r})$ be the spectral sequence of Theorem 3.1. By Proposition
3.4 and the hypotheses that $X(K,f^{h})\neq 0$ and
$\Omega^{n}(K)\neq\emptyset$, the $E_{2}$ page has the form

[title = $E_{\page}$, no y ticks, name = Connected Omega n, cohomological
Serre grading, classes = draw = none , xscale = 2] [background] at (-0.6,0)
n-2; at (-0.6,1) n-1; at (-0.6,2) n;

[”Z_2”](0,2) [”0”](0,1) [”0”](0,0)

[”H^0(Ω^n(K))/Z_2”](1,1) [”H^0(Ω^n-1(K))/Z_2”](1,0) [”0”](1,2)

[”0”](2,1) [”0”](2,0) [”0”](2,2)

$\printpage[name=ConnectedOmegan,page=2]$

and $E_{2}^{i,j}\simeq E_{\infty}^{i,j}$ for all $i,j\in\mathbb{Z}$. So,

$H_{0}(X(K,f^{h}))\simeq\mathbb{Z}_{2}\oplus\left(H^{0}(\Omega^{n}(K))/\mathbb{Z}_{2}\right)\simeq
H^{0}(\Omega^{n}(K)).$

∎

##  6\. Finiteness of Aggregations

In this section, we develop a strategy to reduce sets of aggregations to
finite subsets under the assumption that
$\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})=\emptyset$ and the
spectral curve is smooth and hyperbolic. In particular, we show that under
these assumptions, there is a finite subset $\Lambda_{1}\subseteq\Lambda$ such
that
$\bigcap_{\lambda\in\Lambda}S_{\lambda}=\bigcap_{\lambda\in\Lambda_{1}}S_{\lambda}$.
In order to effectively apply the techniques presented in Section 3.1, we
interpret the necessity of a given aggregation in terms of the emptiness of a
certain system of quadratic equations and inequalities.

See 3.11

###### Proof.

Note that $X(-\mathbb{R}_{+}^{k},f_{k}^{h})\subseteq
X(-\mathbb{R}_{+}^{k+1},f_{k+1}^{h})$. It remains to show the reverse
inclusion. If $f_{\lambda^{(k+1)}}^{h}([x])\not=0$ for all $[x]\in
X(-\mathbb{R}^{k+1}_{+},f_{k+1}^{h})$, then $f_{\lambda^{(k+1)}}^{h}$ has
constant nonzero sign on each connected component of
$X(-\mathbb{R}^{k},f^{h}_{k})$. Since $X(-\mathbb{R}_{+}^{k+1},f_{k+1}^{h})$
has the same number of connected components as
$X(-\mathbb{R}^{k}_{+},f^{h}_{k})$, this implies that
$f_{\lambda^{(k+1)}}^{h}$ is negative on each component of
$X(-\mathbb{R}^{k}_{+},f^{h}_{k})$. So, the equality holds. ∎

######  Remark 6.1.

While Lemma 3.11 is stated projectively, its conclusion implies the affine
statement
$\bigcap_{j=1}^{k}S_{\lambda^{(j)}}=\bigcap_{j=1}^{k+1}S_{\lambda^{(j)}}$. To
see this, note that if $x\in\bigcap_{j=1}^{k}S_{\lambda^{(j)}}$ but $x\not\in
S_{\lambda^{(k+1)}}$, then $f_{\lambda^{(k+1)}}(x)>0$ so that
$f_{\lambda^{(k+1)}}^{h}((x,1))>0$ and therefore $[(x,1)]\in
X(-\mathbb{R}^{k}_{+},f_{k}^{h})\setminus
X(-\mathbb{R}^{k+1}_{+},f_{k+1}^{h})$.

The hypotheses in Lemma 3.11 can be checked by applying the tools in Section
3.1 using the cone $K=-\mathbb{R}^{k}_{+}\times\\{0\\}$.

We apply the construction in Lemma 3.11 specifically to the set of unit length
generators of extreme rays of $\Lambda$ which lie on a proper face of
$\mathbb{R}^{3}_{+}$,

$\Lambda^{\prime}=\\{\lambda\in\mathrm{ex}(\Lambda)\;|\;|\operatorname*{supp}(\lambda)|\leq
2\\}.$

Note that $\Lambda^{\prime}$ has at most 6 elements since each facet of
$\mathbb{R}^{3}_{+}$ can contain at most two extreme rays of $\Lambda$. The
aggregations in $\Lambda^{\prime}$ suffice to describe $\mathop{\rm
conv}(\mathrm{int}(S))$ when the $Q_{i}$ satisfy PDLC by [3, Theorem 2.17]. We
have the following description of elements of $\Lambda^{\prime}$.

######  Proposition 6.2.

If $\lambda\in\Lambda^{\prime}$, then either $\lambda$ is a standard basis
vector of $\mathbb{R}^{3}$ or $g(\lambda)=0$.

###### Proof.

If $\lambda$ is not a standard basis vector, then
$\lambda=\lambda_{i}e_{i}+\lambda_{j}e_{j}$ for some $i\not=j\in[3]$ and
$\lambda_{i},\lambda_{j}>0$. Since $\lambda\in\Lambda^{\prime}$, if
$g(\lambda)\not=0$, then $Q_{\lambda}$ has exactly $n$ positive and one
negative eigenvalue. So, for $\epsilon>0$ sufficiently small,
$\lambda-\epsilon(e_{i}+e_{j})$ and $\lambda+\epsilon(e_{i}+e_{j})$ are
elements of $\mathbb{R}^{3}_{+}$ and $Q_{\lambda\pm\epsilon(e_{i}+e_{j})}$
also has exactly $n$ positive and one negative eigenvalue. But then, $\lambda$
does not span an extreme ray of $\Lambda$, a contradiction.∎

We now work towards applying Lemma 3.11 to the case that
$\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(k)}$ is an enumeration of
$\Lambda^{\prime}$ and $\lambda^{(k+1)}\in\Lambda\setminus\mathop{\rm
cone}(\Lambda^{\prime})$. First, we characterize faces of the relevant cone.

######  Proposition 6.3.

Suppose that $\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(k)}$ is an
enumeration of $\Lambda^{\prime}$ and
$\lambda^{(k+1)}\in\Lambda\setminus\mathop{\rm cone}(\Lambda^{\prime})$. Let
$A\in\mathbb{R}^{(k+1)\times 3}$ have rows
$\left(\lambda^{(i)}\right)^{\top}$. If
$F=\\{t\lambda^{(k+1)}+s\lambda^{(i)}\;|\;i\in[k],t\in\mathbb{R},s\in\mathbb{R}_{+}\\}$
is a face of $A^{\top}(\mathbb{R}^{k}_{+}\times\mathbb{R})$ then
$g(\lambda^{(i)})=0$.

###### Proof.

Note that if $\lambda^{(k+1)}\in\Lambda\setminus\mathop{\rm
cone}(\Lambda^{\prime})$, then $\lambda^{(k+1)}$ has strictly positive
entries. Let $v\in\mathbb{R}^{3}$ support the face $F$ so that
$v^{\top}\lambda^{(k+1)}=v^{\top}\lambda^{(i)}=0$ and
$v^{\top}\lambda^{(j)}>0$ for $j\in[k]\setminus\\{i\\}$. Suppose for the sake
of a contradiction that $g(\lambda^{(i)})\not=0$. By Proposition 6.2, this
implies that $\lambda^{(i)}$ is a standard basis vector of $\mathbb{R}^{3}$.
Relabeling if necessary, take $\lambda^{(i)}=e_{1}$.

Next, note that since $\lambda^{(i)}=e_{1}$ and $g(\lambda^{(i)})\not=0$, the
matrix $Q_{\lambda^{(i)}}=Q_{1}$ has exactly $n$ positive and one negative
eigenvalue. Therefore, either $e_{2}\in\Lambda^{\prime}$ or there is
$t\in[0,1)$ such that $te_{1}+(1-t)e_{2}\in\Lambda^{\prime}$. But then
$v^{\top}e_{2}>0$. Similarly, $v^{\top}e_{3}>0$. Since $\lambda^{(k+1)}$ was
assumed to have strictly positive entries, $v^{\top}\lambda^{(k+1)}>0$, a
contradiction with the construction of $v$. ∎

The following lemmas will also be used in the proof of Theorem 3.12. Both are
concerned with the behavior of aggregations which lie on the oval of depth
$\lfloor\frac{n+1}{2}\rfloor-1$. Let $C$ be the part of the affine cone over
the oval of depth $\lfloor\frac{n+1}{2}\rfloor-1$ where if $\omega\in C$ then
$Q_{\omega}$ has $n-1$ positive eigenvalues, one negative eigenvalue, and 0 as
an eigenvalue of multiplicity one. Lemma 6.4 establishes a root-counting
property. Lemma 6.5 describes the intersection of $C$ with
$\mathbb{R}^{3}_{+}$.

######  Lemma 6.4.

Let $\lambda\in\Lambda\cap C$. Then, if $e\in\mathcal{P}$, there is exactly
one root $t^{*}$ of $g(te+(1-t)\lambda)$ for $t\in(0,1]$ and
$t^{*}e+(1-t^{*})\lambda$ lies on $\partial\mathcal{P}$.

###### Proof.

First, we can reduce to the case that $g$ is a definite representation of the
spectral curve. Indeed, such a definite representation always exists [9] and
the intersection points of the curve with the line
$L=\\{te+(1-t)\lambda\;|\;t\in\mathbb{R}\\}$ do not depend on the
representation of the curve.

Note that there is at least one such root since $e\in\mathcal{P}$. Now, there
are at least $n-1$ roots in $(-\infty,0)\cup(1,\infty)$ since for $t$
sufficiently large, $Q_{\lambda+t(e-\lambda)}$ and $Q_{\lambda-t(e-\lambda)}$
have opposite signature. Since $0$ is a root and there is a root in $(0,1]$,
there can be no other roots, as $g(te+(1-t)\lambda)$ has degree $n+1$.

∎

######  Lemma 6.5.

Suppose that $C\cap\mathbb{R}^{3}_{+}\not=\emptyset$. Then, each connected
component of $C\cap\mathbb{R}^{3}_{+}$ intersects a proper face of
$\mathbb{R}^{3}_{+}$. Moreover, if a connected component of
$C\cap\mathbb{R}^{3}_{+}$ intersects $\mathrm{int}(\mathbb{R}^{3}_{+})$, then
there exist $\lambda^{(1)}\not=\lambda^{(2)}$ in the intersection
$C\cap\mathbb{R}^{3}_{+}$ with $|\operatorname*{supp}(\lambda^{(i)})|\leq 2$.

###### Proof.

Note that the interior of $C$ consists of $\lambda$ such that $Q_{\lambda}$
has $n$ positive and 1 negative eigenvalue. If a connected component $C_{1}$
of $C\cap\mathbb{R}^{3}_{+}$ does not intersect a proper face of
$\mathbb{R}^{3}_{+}$, then $C_{1}$ is contained entirely in
$\mathrm{int}(\mathbb{R}^{3}_{+})$. This implies that the image of $C_{1}$ in
$\mathbb{RP}^{2}$ is an oval of the spectral curve of depth
$\lfloor\frac{n+1}{2}\rfloor-1$. Since there is only one such component, we
have that $C_{1}=C$. Now, the hyperbolicity cone $\mathcal{P}$ is contained in
the interior of the region bounded by $C$ and therefore if $C$ does not
intersect a proper face of $\mathbb{R}^{3}_{+}$, we have that
$\mathcal{P}\subseteq\mathrm{int}(\mathbb{R}^{3}_{+})$. By Proposition 3.4,
this implies that $S=\emptyset$, a contradiction.

The preceding argument shows that $\mathbb{R}^{3}_{+}$ cannot contain the
entirety of the region bounded by $C$. If a connected component $C_{1}$ of
$C\cap\mathbb{R}^{3}_{+}$ contains an element with strictly positive entries
and has a unique element $\lambda$ with
$\left|\operatorname*{supp}(\lambda)\right|\leq 2$, then the entirety of the
region bounded by $C_{1}$ is contained in $\mathbb{R}^{3}_{+}$. In particular,
we have that $C_{1}=C$ and applying the preceding argument gives the desired
contradiction. ∎

We now make the following calculation which will be used in the proof of
Theorem 3.12.

######  Proposition 6.6.

Assume that $C\cap\mathbb{R}^{3}_{+}\not=\emptyset$. Suppose further that
there are $e\in\mathcal{P}$ and $\lambda^{(1)},\lambda^{(2)}\in\Lambda\cap C$
such that $t\mu+(1-t)e=\lambda$ for some $\lambda\in\mathop{\rm
cone}(\lambda^{(1)},\lambda^{(2)})$ and $0\leq t\leq 1$. Set
$A^{\top}=\begin{bmatrix}\lambda^{(1)}&\lambda^{(2)}&\mu\end{bmatrix}$. Then,

$\mathcal{P}\subseteq\mathrm{int}\left(A^{\top}\left(\mathbb{R}^{2}_{+}\times\mathbb{R}\right)\right).$

###### Proof.

We proceed in two steps. First, we show that
$\mathcal{P}\cap\mathrm{int}\left(A^{\top}\left(\mathbb{R}^{2}_{+}\times\mathbb{R}\right)\right)\neq\emptyset$.
Suppose for the sake of a contradiction that
$A^{\top}(\mathbb{R}_{+}^{2}\times\mathbb{R})\cap\mathcal{P}=\emptyset$. Let
$v$ be normal to a separating hyperplane oriented so that $v^{\top}y>0$ for
all $y\in\mathcal{P}$ and $v^{\top}w\leq 0$ for all $w\in
A^{\top}(\mathbb{R}_{+}^{2}\times\mathbb{R})$. Then $v^{\top}\mu<0$ by the
expression $t\mu=\lambda-(1-t)e$. But then, if
$(c_{1},c_{2},d)\in\mathbb{R}^{2}_{+}\times\mathbb{R}$, we have that
$v^{\top}\left(d\mu+\sum_{j=1}^{2}c_{j}\lambda^{(j)}\right)$ can take both
positive and negative values by considering large enough positive and negative
values of $d\in\mathbb{R}$. So, $v$ cannot be the normal to a separating
hyperplane, the desired contradiction.

We now show that
$\\{t\mu+s\lambda^{(i)}\;|\;t\in\mathbb{R},s\in\mathbb{R}_{+}\\}$ does not
intersect $\mathcal{P}$. This follows from Lemma 6.4 since if this
intersection contained some element $e$, then there would be two roots of $g$
restricted to the line segment between $\mu$ and $e$. ∎

The inclusion of cones in the conclusion of Proposition 6.6 implies that
either
$\Omega^{n+1}((A^{\top}(\mathbb{R}^{2}_{+}\times\mathbb{R}))^{\circ})\not=\emptyset$
or that the set
$\Omega^{n}((A^{\top}(\mathbb{R}^{2}_{+}\times\mathbb{R}))^{\circ})$ has
nontrivial first cohomology group. This calculation is leveraged to prove
Theorem 3.12.

See 3.12

###### Proof.

Enumerate
$\Lambda^{\prime}=\\{\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(k)}\\}$ and
let $\lambda^{(k+1)}\in\Lambda\setminus\mathop{\rm cone}(\Lambda^{\prime})$.
We will show that $\lambda^{(k+1)}$ is not necessary to describe the set
defined by all aggregations by showing that
$\bigcap_{i=1}^{k}S_{\lambda^{(i)}}=\bigcap_{i=1}^{k+1}S_{\lambda^{(i)}}$.
Note that if $g(\lambda^{(k+1)})\not=0$, then we can write $\lambda^{(k+1)}$
as a conical combination of an element of $\mathop{\rm
cone}(\Lambda^{\prime})$ and an element of $C$, where $C$ is the portion of
the affine cone over the oval of depth $\lfloor\frac{n+1}{2}\rfloor-1$ which
contains matrices with one negative eigenvalue. So, it suffices to take
$\lambda^{(k+1)}\in C$. By the hypothesis that $S\not=\emptyset$, the
hyperbolicity cone $\mathcal{P}\not\subseteq\mathbb{R}^{3}_{+}$. This implies
that there is $e\in\mathcal{P}$ and $\lambda^{(i)},\lambda^{(j)}\in\Lambda$
such that $t\lambda^{(k)}+(1-t)e=\lambda$ for some $\lambda\in\mathop{\rm
cone}(\lambda^{(i)},\lambda^{(j)})$ and $0\leq t\leq 1$. Since
$\lambda^{(k+1)}\not\in\mathop{\rm cone}(\Lambda^{\prime})$, we can take
$\lambda^{(i)},\lambda^{(j)}\in C$.

Set $K=\mathop{\rm cone}(\lambda^{(i)},\lambda^{(j)},\mu,-\mu)$. By
Proposition 6.6, $\mathcal{P}\subseteq\mathrm{int}(K^{\circ})$ and therefore
either $H^{1}(\Omega^{n}(K^{\circ}))\not=0$ or
$\Omega^{n+1}(K^{\circ})\neq\emptyset$. By Proposition 3.4, this implies that
$X(K^{\circ},f^{h})=\emptyset$, i.e.
$\mathcal{V}_{\mathbb{R}}(f_{\lambda^{(k+1)}}^{h})\cap
X(-\mathbb{R}^{2}_{+},(f_{\lambda^{(i)}}^{h},f_{\lambda^{(j)}}^{h}))=\emptyset$.
This in turn implies that
$\mathcal{V}_{\mathbb{R}}(f_{\lambda^{(k+1)}}^{h})\cap X(\mathop{\rm
cone}(\Lambda^{\prime})^{\circ},f^{h})=\emptyset$. By the hypothesis that
$\Omega^{n}(\mathop{\rm cone}(\Lambda^{\prime})^{\circ})$ is contractible and
Lemma 3.11, this implies that $\lambda^{(k+1)}$ is unnecessary, so that
$\bigcap_{i=1}^{k}S_{\lambda^{(i)}}=\bigcap_{i=1}^{k+1}S_{\lambda^{(i)}}$. ∎

Figure 5 illustrates the argument of the proof of Theorem 3.12. Again, we take
the system of quadratics from Example 3.8 so that the curve
$\mathcal{V}_{\mathbb{R}}(g)$ is hyperbolic but $g$ is not a definite
representation. The aggregation $\lambda^{(k+1)}$ which lies in the interior
of $\mathbb{R}^{3}$ does not contribute to the intersection of all
aggregations of correct signature, which is given by
$\bigcap_{\lambda\in\Lambda^{\prime}}S_{\lambda}$.

Figure 5. An illustration of Theorem 3.12. The set described by aggregations
of the correct signature is $\bigcap_{\lambda\in\Lambda^{\prime}}S_{\lambda}$.
The aggregation $\mu$ does not contribute to this description. This is
certified by the fact that the cone
$K=A^{\top}(\mathbb{R}^{4}_{+}\times\mathbb{R})$ contains the hyperbolicity
cone of $g$ in its interior. In cohomological terms,
$H^{1}(\Omega^{n}(K^{\circ}))=\mathbb{Z}_{2}$.

As shown in Example 3.13, the set defined by the intersection of all
permissible aggregations is not necessarily defined by the intersection of
aggregations in $\Lambda^{\prime}$ when the assumption on the contractibility
of $\Omega^{n}(\mathop{\rm cone}(\Lambda^{\prime})^{\circ})$ is not met.
However, we are able to develop a bound on the number of aggregations with
positive entries needed when PDLC is not satisfied and the set defined by all
permissible aggregations is not connected. First, we observe that adding an
aggregation with strictly positive entries to $\Lambda^{\prime}$ cannot make
the number of connected components increase.

######  Lemma 6.7.

Set $K_{0}=\mathop{\rm cone}(\Lambda^{\prime})$ and let
$\mu^{(1)},\mu^{(2)},\ldots,\mu^{(N)}\in\Lambda$ have strictly positive
entries. For each $1\leq j\leq N$, set $K_{j}=\mathop{\rm
cone}(\Lambda^{\prime}\cup\bigcup_{i=1}^{j}\\{\mu^{(i)}\\})$. Then, every
connected component of $\Omega^{n}(K_{j}^{\circ})$ intersects a component of
$\Omega^{n}(K_{0}^{\circ})$. Moreover,

$1\leq\dim H^{0}(\Omega^{n}(K_{N}^{\circ}))\leq\dim
H^{0}(\Omega^{n}(K_{N-1}^{\circ}))\leq\dots\leq\dim
H^{0}(\Omega^{n}(K_{1}^{\circ}))\leq\dim H^{0}(\Omega^{n}(K_{0}^{\circ})).$

###### Proof.

We first show that there are no connected components of
$\Omega^{n}(K_{j}^{\circ})$ where the points contain all positive entries.
Suppose for the sake of a contradiction that there was such a connected
component. Then, its boundary would form an oval of
$\mathcal{V}_{\mathbb{R}}(g)$ whose interior contains matrices with $n$
positive eigenvalues. If
$\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})=\emptyset$, then
$\mathcal{V}_{\mathbb{R}}(g)$ is hyperbolic and the only oval which contains
matrices with $n$ positive eigenvalues in its interior is the oval of depth
$\lfloor\frac{n+1}{2}\rfloor-1$. So, if $K_{j}$ contains this oval, it
contains the oval of maximum depth and therefore the hyperbolicity cone of $g$
which consists of either positive definite matrices or matrices with exactly
two negative eigenvalues. By Proposition 3.4, this certifies that
$X(K_{j}^{\circ},f^{h})=\emptyset$. This is a contradiction since
$S\not=\emptyset$ and therefore $X(-\mathbb{R}^{3}_{+},f^{h})\not=\emptyset$
but $X(-\mathbb{R}^{3}_{+},f^{h})\subseteq X(K_{j}^{\circ},f^{h})$.

This implies that every connected component of $\Omega^{n}(K_{j}^{\circ})$ has
nonempty intersection with $\Omega^{n}(K_{0}^{\circ})$. Indeed, there is
$\lambda$ in this component with $\lambda_{i}=0$ for some $i\in[3]$. Therefore
either $\lambda\in\Lambda^{\prime}$ or $\lambda$ is a conical combination of
two elements of $\Lambda^{\prime}$. Note that since
$\Omega^{n}(K_{0}^{\circ})\subseteq\Omega^{n}(K_{j-1}^{\circ})\subseteq\Omega^{n}(K_{j}^{\circ})$
for all $j$, this implies that each connected component of
$\Omega^{n}(K_{j}^{\circ})$ intersects $\Omega^{n}(K_{j-1}^{\circ})$ .

The last claim follows since
$\Omega^{n}(K_{j-1}^{\circ})\subseteq\Omega^{n}(K_{j}^{\circ})$ for each $j$.
So, if each connected component of $\Omega^{n}(K_{j}^{\circ})$ intersects
$\Omega^{n}(K_{j-1}^{\circ})$, then there are at least as many connected
components of $\Omega^{n}(K_{j-1}^{\circ})$ as connected components of
$\Omega^{n}(K_{j}^{\circ})$. Finally, $\dim
H^{0}(\Omega^{n}(K^{\circ}_{N}))\not=0$ by Theorem 3.1 since
$\emptyset\not=X(-\mathbb{R}^{3}_{+},f^{h})\subseteq X(K_{N}^{\circ},f^{h})$
and therefore $\dim H_{0}(X(K_{N}^{\circ},f^{h}))\geq 1$. ∎

We can now prove Theorem 3.14.

See 3.14

###### Proof.

By Lemma 3.11 and Proposition 6.6, given some finite list of aggregations
$\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(k)}$ and the cone
$K_{0}=\mathop{\rm cone}(\\{\lambda^{(i)}\;|\;i\in[k]\\})$, a sequence of
aggregations $\mu^{(1)},\mu^{(2)},\ldots,\mu^{(N)}\in\Lambda$, each with
positive entries, contains only necessary elements only if $\dim
H^{0}(\Omega^{n}(K_{j}^{\circ}))\not=\dim H^{0}(\Omega^{n}(K_{j-1}^{\circ}))$,
where $K_{j}=\mathop{\rm
cone}(K_{0}\cup\bigcup_{\ell=1}^{j}\\{\mu^{(\ell)}\\})$. By Lemma 6.7, this
only happens if $\dim H^{0}(\Omega^{n}(K_{j})^{\circ})<\dim
H^{0}(\Omega^{n}(K_{j-1}^{\circ}))$. In particular, the maximal length $N$ of
a sequence of aggregations with positive entries such that $\dim
H^{0}(\Omega^{n}(K_{j}^{\circ}))<\dim H^{0}(\Omega^{n}(K_{j-1}^{\circ}))$ for
all $1\leq j\leq N$, starting with $K_{0}=\mathop{\rm
cone}(\Lambda^{\prime})$, is at most $\dim H^{0}(\Omega^{n}(\mathop{\rm
cone}(\Lambda^{\prime})^{\circ}))-1$. ∎

Together, Theorems 3.12 and 3.14 imply Theorem 1.3.

See 1.3

###### Proof.

If $\Omega^{n}(\mathop{\rm cone}(\Lambda^{\prime})^{\circ})$ is contractible,
then $\Lambda_{1}=\Lambda^{\prime}$ by Theorem 3.12. Otherwise, the result is
Theorem 3.14. ∎

##  7\. A Sufficient Condition for Obtaining the Convex Hull via Aggregations

In this section, we present proofs of the results stated in Section 3.2.
Throughout this section, we assume that
$\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})=\emptyset$. Our
approach relies on certifying that $S^{h}\cap H=0$ using positive definite
aggregations of the restrictions $Q_{i}|_{H}$.

We first prove a variant of [6, Lemma 5.4] that is compatible with the closed
inequality and projective setting.

######  Lemma 7.1.

Suppose that
$S^{h}\cap\\{(x,x_{n+1})\in\mathbb{R}^{n+1}\;|\;x_{n+1}=0\\}=\\{0\\}$. If
$\alpha\in\mathbb{R}^{n}$ and $\beta\in\mathbb{R}$ are such that
$\alpha^{\top}x<\beta$ for all $x\in S$ and if
$H=\\{(u,u_{n+1})\in\mathbb{R}^{n+1}\;|\;\alpha^{\top}u=\beta u_{n+1}\\}$,
then $S^{h}\cap H=0$.

###### Proof.

If $0\not=(\hat{u},\hat{u}_{n+1})\in S^{h}\cap H$, then
$\frac{\hat{u}}{|\hat{u}_{n+1}|}\in S$ and
$\alpha^{\top}\left(\frac{\hat{u}}{\hat{u}_{n+1}}\right)=\beta$. ∎

We now record a variant of the proof strategy of [6, Theorem 2.4] and [3,
Theorem 2.8].

See 3.5

###### Proof.

Suppose that $y\not\in\mathop{\overline{\rm conv}}(S)$. Then, there is
$\alpha\in\mathbb{R}^{n}$ such that $\alpha^{\top}x<\alpha^{\top}y$ for all
$x\in S$. Let $\beta=\alpha^{\top}y$ and
$H=\\{(x,x_{n+1})\;|\;\alpha^{\top}x=\beta x_{n+1}\\}$. If property (* ‣ 3.5)
holds, then there is $\lambda\in\mathbb{R}^{3}_{+}$ such that
$Q_{\lambda}|_{H}\succ 0$. By the Cauchy Interlacing Theorem and the
hypothesis that $\mathrm{int}(S)\not=\emptyset$, it follows that $Q_{\lambda}$
has exactly one negative eigenvalue. Since $Q_{\lambda}|_{H}\succ 0$, this in
turn implies that either $S_{\lambda}$ is convex or that $S_{\lambda}$
consists of two disjoint convex connected components separated by the affine
hyperplane $\\{x\in\mathbb{R}^{n}\;|\;\alpha^{\top}x=\beta\\}$. Since
$\alpha^{\top}x<\beta$ is valid on $S$, this implies that
$\mathop{\overline{\rm conv}}(S)$ is contained in a single convex connected
component of $S_{\lambda}$ and therefore $\lambda$ is a good aggregation which
certifies that $y\not\in\mathop{\overline{\rm conv}}(S)$. ∎

In light of Proposition 3.5, any condition which ensures that the emptiness of
$S^{h}\cap H$ is certified by positive definite aggregations of the
$Q_{i}|_{H}$ also ensures that $\mathop{\overline{\rm conv}}(S)$ has a
representation as the intersection of good aggregations.

We first restrict our attention to the case that the polynomials
$g(\lambda)=\det(Q_{\lambda})$ and $g_{H}(\lambda)=\det(Q_{\lambda}|_{H})$ are
both smooth and hyperbolic. The first step is to understand the relationship
between the curves $\mathcal{V}_{\mathbb{R}}(g)$ and
$\mathcal{V}_{\mathbb{R}}(g_{H})$. In the case that $Q_{1},Q_{2},Q_{3}$
satisfy PDLC, then $g_{H}$ interlaces $g$ [12]. In the case that
$\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})=\emptyset$ but the
$Q_{i}$ do not satisfy PDLC, the relationship is more subtle. If $g$ is
hyperbolic, we set $\mathcal{P}$ to be the hyperbolicity cone of $g$ which
contains either positive definite matrices or matrices with exactly two
negative eigenvalues.

If $\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})=\emptyset$, then
$\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})\cap H=\emptyset$.
Theorem 3.3 then implies that a hyperbolicity cone $\mathcal{P}_{H}$ of
$g_{H}$ contains either positive definite matrices or matrices with exactly
two negative eigenvalues. We apply interlacing properties of eigenvalues to
understand how the ovals of $\mathcal{V}_{\mathbb{R}}(g)$ and
$\mathcal{V}_{\mathbb{R}}(g_{H})$ interact by restricting to a pencil (see
also [14, 15]).

######  Lemma 7.2.

Suppose that $\lambda^{(1)},\lambda^{(2)}\in\mathbb{R}^{3}$ are such that
$g(\lambda^{(i)})=0$ for $i=1,2$ and that
$g(t\lambda^{(1)}+(1-t)\lambda^{(2)})\not=0$ for $t\in(0,1)$. Suppose further
that $g_{H}(\lambda^{(i)})\not=0$ for $i=1,2$. The number of zeros of
$g_{H}(t\lambda^{(1)}+(1-t)\lambda^{(2)})$ for $t\in(0,1)$ is even if
$Q_{\lambda^{(1)}}$ and $Q_{\lambda^{(2)}}$ have the same signature and odd
otherwise.

###### Proof.

Let $r\in\mathbb{Z}$ be such that $Q_{\lambda^{(1)}}$ has $r$ positive
eigenvalues, $n-r$ negative eigenvalues, and 0 as an eigenvalue of
multiplicity one. Then, $Q_{\lambda^{(1)}}|_{H}$ also has $r$ positive
eigenvalues and $n-r$ negative eigenvalues by the Cauchy Interlacing Theorem.

If $Q_{\lambda^{(1)}}$ and $Q_{\lambda^{(2)}}$ have the same signature, then
$Q_{\lambda^{(2)}}|_{H}$ also has $r$ positive and $n-r$ negative eigenvalues.
In particular, $g_{H}(\lambda^{(1)})$ and $g_{H}(\lambda^{(2)})$ have the same
sign which implies that $g_{H}(t\lambda^{(1)}+(1-t)\lambda^{(2)})=0$ for an
even number of $t\in(0,1)$.

On the other hand, if $Q_{\lambda^{(2)}}$ has $r+1$ positive eigenvalues,
$n-r-1$ negative eigenvalues, and 0 as an eigenvalue of multiplicity one, then
$Q_{\lambda^{(2)}}|_{H}$ has $r+1$ positive eigenvalues and $n-r-1$ negative
eigenvalues. In particular, $g_{H}(\lambda^{(1)})$ and $g_{H}(\lambda^{(2)})$
have different signs and therefore $g_{H}(t\lambda^{(1)}+(1-t)\lambda^{(2)})$
for an odd number of $t\in(0,1)$. The case where $Q_{\lambda^{(2)}}$ has $r-1$
positive and $n-r+1$ negative eigenvalues is similar. ∎

Lemma 7.2 reduces the possible interactions between
$\mathcal{V}_{\mathbb{R}}(g)$ and $\mathcal{V}_{\mathbb{R}}(g_{H})$. In order
to prove Theorem 3.6, we use the following spectral sequence from [1] which
computes relative homology groups of hyperplane sections of the solution set
to a system of quadratic inequalities.

######  Theorem 7.3 ([1, Theorem D]).

Fix a polyhedral cone $K\subseteq\mathbb{R}^{m}$, a homogenous quadratic map
$p:\mathbb{R}^{n+1}\to\mathbb{R}^{m}$, and a hyperplane
$\bar{H}\subseteq\mathbb{RP}^{n}$. There is a first quadrant cohomology
spectral sequence $(G_{r},d_{r})$ converging to
$H_{n-*}(X(K,p),X(K,p)\cap\bar{H})$ with

$G_{2}^{i,j}=H^{i}(\Omega^{j}_{H}(K),\Omega^{j+1}(K))\text{ for }j>0,\quad
G_{2}^{i,0}=H^{i}(C\Omega(K),\Omega^{1}(K)).$

Applying Theorem 7.3 to hyperplane sections of an empty variety gives the
following

######  Corollary 7.4.

Fix a hyperplane $H\subseteq\mathbb{R}^{n+1}$. Suppose that
$\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})=\emptyset$, and that
the polynomial $g$ is smooth and hyperbolic, and that $\mathcal{P}$ does not
contain a positive definite matrix. Then,
$H^{1}(\Omega^{n-1}_{H}(\\{0\\}),\Omega^{n}(\\{0\\}))=0$. That is, every
noncontractible loop in $\Omega^{n-1}_{H}(\\{0\\})$ can be deformed to a
noncontractible loop in $\Omega^{n}(\\{0\\})$.

###### Proof.

Let $(G_{r},d_{r})$ be the spectral sequence of Theorem 7.3. Set
$X=\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})$,
$\Omega^{j}_{H}=\Omega^{j}_{H}(\\{0\\})$, and $\Omega^{j}=\Omega^{j}(\\{0\\})$
for notational simplicity.

First suppose that $n=2$. We then have that
$G_{\infty}^{1,1}\simeq\ker(d_{2}^{1,1}:G_{2}^{1,1}\to G_{2}^{3,0})$. Since
$X$ and $X\cap\bar{H}$ are empty,

$0=H_{0}(X,X\cap\bar{H})\simeq G_{\infty}^{2,0}\oplus G_{\infty}^{1,1}\oplus
G_{\infty}^{0,2}.$

Therefore, $G_{\infty}^{1,1}\simeq 0$ and $d_{2}^{1,1}$ is injective. But
$G_{2}^{3,0}=H^{3}(C\Omega,\Omega^{1})=H^{3}(B^{3},\mathbb{S}^{2})=0$ and
therefore $G_{2}^{1,1}=H^{1}(\Omega_{H}^{1},\Omega^{2})=0$.

Now suppose that $n\geq 3$. We first show that $G^{i,j}_{2}=0$ when $i\geq 3$
and $j\geq 1$. From the long exact sequence of the pair
$(\Omega^{j}_{H},\Omega^{j+1})$, we get the following exact sequence

$\cdots\to H^{i-1}(\Omega^{j+1})\to H^{i}(\Omega_{H}^{j},\Omega^{j+1})\to
H^{i}(\Omega_{H}^{j})=0\to\cdots,$

where $H^{i}(\Omega_{H}^{j})=0$ for $i\geq 3$. So it suffices to show that
$H^{i-1}(\Omega^{j+1})=0$ for $i\geq 3$ and $j\geq 1$. By Theorem 3.3 and the
hypothesis that $\mathcal{P}$ contains matrices with exactly two negative
eigenvalues, we compute that if $n\geq 4$, then $\Omega^{j+1}$ is homotopy
equivalent to the union of two points when $j=1$, homotopy equivalent to a
point when $2\leq j\leq n-2$, homotopy equivalent to $\mathbb{S}^{1}$ when
$j=n-1$, and empty when $j\geq n$. If $n=3$, $\Omega^{1+1}$ is homotopy
equivalent to the union of two points, $\Omega^{2+1}$ is homotopy equivalent
to $\mathbb{S}^{1}$, and $\Omega^{3+1}=\emptyset$.

In all cases, we see that $H^{i-1}(\Omega^{j+1})=0$ when $i\geq 3$ so that

$0\to H^{i}(\Omega^{i}_{H},\Omega^{j+1})\to 0$

is exact. So, $G^{i,j}_{2}=H^{i}(\Omega^{i}_{H},\Omega^{j+1})=0$ when $i\geq
3$ and $j\geq 1$.

Next, since $\mathop{\rm span}\nolimits\\{Q_{1},Q_{2},Q_{3}\\}$ does not
contain a positive semidefinite matrix, it does not contain a negative
semidefinite matrix so that $\Omega^{1}=\mathbb{S}^{2}$. This implies that
$G^{i,0}=H^{i}(C\Omega,\Omega^{1})=H^{i}(B^{3},\mathbb{S}^{2})=0$ for $i\geq
4$.

The vanishing of $G_{2}^{i,j}$ for $i\geq 3,j\geq 1$ and $i\geq 4$ and the
hypothesis that $n\geq 3$ imply that $H^{1}(\Omega^{n-1}_{H},\Omega^{n})\simeq
G_{2}^{1,n-1}\simeq G_{\infty}^{1,n-1}$. Since $X$ and $X\cap\bar{H}$ are
empty, and since

$0=H_{0}(X,X\cap\bar{H})\simeq\bigoplus_{j=0}^{n}G^{n-j,j}_{\infty}$

by Theorem 7.3, $H^{1}(\Omega^{n-1}_{H},\Omega^{n})=0$. ∎

We are now able to prove Theorem 3.6.

See 3.6

###### Proof.

Suppose that $\mathcal{P}_{H}$ does not contain positive definite matrices. By
Theorem 3.3, if $\lambda\in\mathcal{P}_{H}$ then $Q_{\lambda}|_{H}$ has
exactly two negative eigenvalues and $n-2$ positive eigenvalues. Suppose for
the sake of a contradiction that $\mathcal{P}_{H}\not\subseteq\mathcal{P}$.

Let $e\in\mathcal{P}$ and $a\in\mathbb{R}^{3}$ such that the line
$\ell=\\{te+(1-t)a\;|\;t\in\mathbb{R}\\}\subseteq\mathbb{R}^{3}$ has
$\ell\cap\mathcal{V}_{\mathbb{R}}(g)\cap\mathcal{V}_{\mathbb{R}}(g_{H})=\emptyset$
and the intersections $\ell\cap\mathcal{V}_{\mathbb{R}}(g)$ and
$\ell\cap\mathcal{V}_{\mathbb{R}}(g_{H})$ are transverse. This is possible
since $\mathcal{V}_{\mathbb{R}}(g)\cap\mathcal{V}_{\mathbb{R}}(g_{H})$ has at
most finitely many points. By hyperbolicity of $g$, the polynomial
$g(te+(1-t)a)$ has roots $t_{1}<t_{2}<\ldots<t_{n+1}$. Let $2\leq j\leq n-1$
be such that $t_{j}<1<t_{j+1}$. Then, $g_{H}(te+(1-t)a)$ has roots
$s_{1}<s_{2}<\ldots<s_{n}$. By Lemma 7.2 and the Cauchy Interlacing Theorem,
there are either three roots $s_{\ell}<s_{\ell+1}<s_{\ell+2}$ in between
$t_{j-2}$ and $t_{j-1}$ or between $t_{j+2}$ and $t_{j+3}$ or that there are
two roots $t_{j}<s_{\ell}<s_{\ell+1}<t_{j+1},$ where $t_{j-2}=-\infty$ if
$j-2=0$ and $t_{j+3}=\infty$ if $j+3=n+2$. By the hypothesis that
$\mathcal{P}_{H}\not\subseteq\mathcal{P}$, the last case is impossible.

In the first two cases, the image of $\mathcal{P}_{H}\cap\mathbb{S}^{2}$ in
$\mathbb{RP}^{2}$ is contained in the interior of the oval of
$\mathcal{V}_{\mathbb{R}}(g)$ of depth $\lfloor\frac{n+1}{2}\rfloor-2$ and the
exterior of the oval of $\mathcal{V}_{\mathbb{R}}(g)$ of depth
$\lfloor\frac{n+1}{2}\rfloor-1$. Note that the image of $\Omega^{n}$ in
$\mathbb{RP}^{2}$ is the intersection of the interior of the oval of
$\mathcal{V}_{\mathbb{R}}(g)$ of depth $\lfloor\frac{n+1}{2}\rfloor-1$ and the
exterior of the oval of $\mathcal{V}_{\mathbb{R}}(g)$ of depth
$\lfloor\frac{n+1}{2}\rfloor$.

Now, the images of $\mathcal{P}_{H}\cap\mathbb{S}^{2}$ and
$\mathcal{P}\cap\mathbb{S}^{2}$ in $\mathbb{RP}^{2}$ are bounded by two
disjoint ovals, neither of which contains the other. Let $\sigma$ be a
representative of a nontrivial class in $H_{1}(\Omega_{H}^{n})\simeq
H^{1}(\Omega_{H}^{n})$. By the above observation, $\sigma$ can be chosen so
that its image in $\mathbb{RP}^{2}$ intersects the image of $\Omega^{n}$ and
therefore gives a nontrivial class in
$H_{1}(\Omega_{H}^{n-1},\Omega^{n})\simeq H^{1}(\Omega_{H}^{n-1},\Omega^{n})$,
contradicting Corollary 7.4. ∎

Figure 6 shows the two possible containment patterns for $g$ and $g_{H}$ on
the curve given in Example 3.8.

Figure 6. The two possible containment patterns for $\mathcal{P}$ and
$\mathcal{P}_{H}$ where $g$ is given as in Example 3.8. On the left,
$\mathcal{P}_{H}\subseteq\mathcal{P}$ and the emptiness of
$\mathcal{V}_{\mathbb{R}}(f^{h}_{1},f^{h}_{2},f^{h}_{3})\cap H$ is certified
by $Q_{\lambda}|_{H}$ having two negative eigenvalues for
$\lambda\in\mathcal{P}_{H}$ and $H^{1}(\Omega^{n-1}_{H})\not=0$. On the right,
$\mathcal{P}_{H}\not\subseteq\mathcal{P}$ and $Q_{\lambda}|_{H}\succ 0$ for
$\lambda\in\mathcal{P}_{H}$.

We now combine the ideas of Theorem 3.6 and Proposition 3.5 to obtain a
sufficient condition for describing $\mathop{\overline{\rm conv}}(S)$ via
aggregations.

See 3.7

###### Proof.

It suffices to show that these hypotheses ensure that (* ‣ 3.5) is satisfied.
If $\lambda\in\mathcal{P}$ implies that $Q_{\lambda}\succ 0$, then the result
is [6, Theorem 2.4]. Otherwise, let $H$ be a hyperplane such that $S^{h}\cap
H=0$. This implies that the set $X(K,f^{h})\cap\bar{H}=\emptyset$ where
$K=-\mathbb{R}^{3}_{+}$ is the nonpositive orthant. By Proposition 3.4, either
there is $\lambda\in\mathbb{R}^{3}_{+}$ such that $Q_{\lambda}|_{H}\succ 0$ or
$H^{1}(\Omega^{n-1}_{H}(K))\neq 0$. In either case, $g_{H}$ is hyperbolic if
it is smooth. However, since $\Omega^{n-1}(K)\subseteq
K^{\circ}\cap\mathbb{S}^{2}$, a nontrivial group $H^{1}(\Omega^{n-1}_{H}(K))$
would imply that $\mathcal{P}_{H}\subseteq K^{\circ}=\mathbb{R}^{3}_{+}$. On
the other hand, by the assumption that
$\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})=\emptyset$ and
Theorem 3.6, $\mathcal{P}_{H}\subseteq\mathcal{P}$. Since
$\mathcal{P}\cap\mathbb{R}^{3}_{+}=\emptyset$, this therefore implies that
$H^{1}(\Omega^{n}(K))=0$.

If $g_{H}$ is not smooth and $H^{1}(\Omega^{n-1}_{H}(K))\neq 0$, then by the
Cauchy Interlacing Theorem and since $\mathcal{P}\cap\mathbb{R}^{3}_{+}=0$,
there is $\lambda\in K^{\circ}$ such that $[\lambda]\in\mathbb{RP}^{2}$ lies
on the outside of the oval $\mathcal{V}_{\mathbb{R}}(g)$ of depth
$\lfloor\frac{n+1}{2}\rfloor-1$ and such that $Q_{\lambda}|_{H}$ has at most
$n-2$ positive eigenvalues. But then, a nontrivial class in
$H^{1}(\Omega^{n-1}_{H}(K))$ gives a nontrivial class of
$H^{1}(\Omega^{n-1}_{H}(K),\Omega^{n}(K))$, contradicting Corollary 7.4.

So, there is $\lambda\in\mathbb{R}^{3}_{+}$ with $Q_{\lambda}|_{H}\succeq 0$
and therefore (* ‣ 3.5) is satisfied. ∎

######  Remark 7.5.

Under the conditions of Theorem 3.7, the assumption that $S$ has no points at
infinity can be certified via aggregations. This follows by applying Theorem
3.6 with $H=\\{(x,x_{n+1})\in\mathbb{R}^{n+1}\;|\;x_{n+1}=0\\}$, as the
hypothesis that $\mathcal{P}\cap\mathbb{R}^{3}_{+}=\emptyset$ implies that the
certificate that $S^{h}\cap H=0$ cannot come from matrices with two negative
eigenvalues.

We conclude this section by proving Theorem 1.2.

See 1.2

###### Proof.

The statement that $\mathop{\overline{\rm conv}}(S)$ is given by the
intersection of good aggregations when $g$ is hyperbolic is Theorem 3.7. In
the $n=2$ case when $g$ is not hyperbolic, note that condition (* ‣ 3.5) is
satisfied: A nontrivial class in $H^{1}(\Omega_{H}^{1}(K))$ would give a
nontrivial class in $H^{1}(\Omega_{H}^{1}(K),\Omega^{2}(K))$ in this case,
contradicting Corollary 7.4. Therefore by Proposition 3.4, there is
$\lambda\in\mathbb{R}^{3}_{+}$ such that $Q_{\lambda}|_{H}\succ 0$.

It remains to show that when the spectral curve is hyperbolic, a finite number
of good aggregations recovers $\mathop{\overline{\rm conv}}(S)$. For each
$y\in\mathbb{R}^{n}\setminus\mathop{\overline{\rm conv}}(S)$, let
$\lambda^{(y)}\in\mathbb{R}^{3}$ be a good aggregation which certifies that
$y\not\in\mathop{\overline{\rm conv}}(S)$ and set
$\Lambda_{1}=\\{\lambda^{(y)}\;|\;y\in\mathbb{R}^{n}\setminus\mathop{\overline{\rm
conv}}(S)\\}$ to be the set of all such $\lambda^{(y)}$. Now, if
$\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(k)}$ are the unit length
generators of the extreme rays of $\Lambda_{1}$ with
$|\operatorname*{supp}(\lambda^{(i)})|\leq 2$, then
$\bigcap_{i=1}^{k}S_{\lambda^{(i)}}=\bigcap_{\lambda\in\Lambda_{1}}S_{\lambda}$
by Lemma 3.11 and Propositions 6.6 and 3.4, as in the proof of Theorem 3.12.
Note that $k\leq 6$ since each $\lambda^{(i)}$ has
$|\operatorname*{supp}(\lambda^{(i)})|\leq 2$ and generates an extreme ray of
$\Lambda_{1}$. ∎

##  8\. Topology of the Set of Permissible Aggregations

In this section we provide proofs of the statements in Section 3.4. Throughout
this section, we assume that $\mathrm{int}(S)\not=\emptyset$. Recall that
$\Lambda=\\{\lambda\in\mathbb{R}^{3}_{+}\;|\;Q_{\lambda}\text{ has exactly one
negative eigenvalue}\\}$ and that a permissible aggregation
$\lambda\in\Lambda$ is a _good aggregation_ if $\mathop{\rm conv}(S)\subseteq
S_{\lambda}$ and a _bad aggregation_ otherwise.

See 3.15

###### Proof.

We first show that the set of good aggregations is closed in $\Lambda$. Let
$\lambda^{(i)}\to\lambda$ be a sequence of aggregations which converges to
$\lambda\in\Lambda$. Recall that each matrix $Q_{\lambda}$ has block strucutre
$Q_{\lambda}=\begin{bmatrix}A_{\lambda}&b_{\lambda}\\\
b_{\lambda}^{\top}&c_{\lambda}\end{bmatrix}$, where
$A_{\lambda}\in\mathbb{R}^{n\times n}$ defines the homogeneous part of
$f_{\lambda}$. If $\mathrm{int}(S_{\lambda^{(i)}})$ is convex for sufficiently
large $i$, then $A_{\lambda^{(i)}}\succeq 0$ for sufficiently large $i$. Since
the cone of positive semidefinite matrices is closed, $A_{\lambda}\succeq 0$
so that $\mathrm{int}(S_{\lambda})$ is convex and therefore $\lambda$ is a
good aggregation. Otherwise, suppose that $\mathrm{int}(S_{\lambda^{(i)}})$
has two convex components for large $i$. Let
$(\alpha_{i},\beta_{i})\in\mathbb{R}^{n+1}$ be the unit length eigenvector
corresponding to the negative eigenvalue of $Q_{\lambda^{(i)}}$ with sign
chosen such that $\alpha_{i}^{\top}x\leq\beta_{i}$ for all $x\in S$. This is
possible since each $\lambda^{(i)}$ is a good aggregation and the affine
hyperplane defined by $\alpha_{i}^{\top}x=\beta_{i}$ separates the two
components of $\mathrm{int}(S_{\lambda^{(i)}})$. Now, $\alpha_{i}\to\alpha$
and $\beta_{i}\to\beta$ where $(\alpha,\beta)$ is an eigenvector of
$Q_{\lambda}$ corresponding to the negative eigenvalue and
$\alpha^{\top}x\leq\beta$ for all $x\in S$. So, $\lambda$ is a good
aggregation.

On the other hand, the set of bad aggregations is also closed in $\Lambda$.
Let $\lambda^{(i)}$ be a sequence of aggregations which converges to
$\lambda\in\Lambda$. Let $(\alpha_{i},\beta_{i})\in\mathbb{R}^{n+1}$ be the
unit length eigenvector of $Q_{\lambda^{(i)}}$ corresponding to the negative
eigenvalue of $Q_{\lambda^{(i)}}$ oriented so that $\alpha_{i}\to\alpha$ and
$\beta_{i}\to\beta$ where $(\alpha,\beta)$ is an eigenvector of $Q_{\lambda}$
corresponding to the negative eigenvalue. Since each $\lambda^{(i)}$ is a bad
aggregation, there are $x,y\in\mathrm{int}(S)$ such that
$(\alpha^{(i)})^{\top}x<\beta^{(i)}$ and $(\alpha^{(i)})^{\top}y>\beta^{(i)}$
for sufficiently large $i$. This implies that $\alpha^{\top}x\leq\beta$ and
$\alpha^{\top}y\geq\beta$. Since $x,y\in\mathrm{int}(S)$, this implies the
existence of $\tilde{x},\tilde{y}\in\mathrm{int}(S)$ such that
$\alpha^{\top}\tilde{x}<\beta$ and $\alpha^{\top}\tilde{y}>\beta$. Therefore
$\lambda$ is a bad aggregation.

If $\Lambda_{1}$ is a connected component of $\Lambda$, then we can write
$\Lambda_{1}$ as the disjoint union of good aggregations in $\Lambda_{1}$ and
bad aggregations in $\Lambda_{1}$. Since $\Lambda_{1}$ is connected, this
implies that $\Lambda_{1}$ consists entirely of good aggregations or entirely
of bad aggregations. ∎

######  Remark 8.1.

Note that Theorem 1.2 implies that if PDLC is satisfied and $S$ has no points
at infinity, then non-intersection of $\mathop{\overline{\rm conv}}(S)$ with
an affine hyperplane $H$ is can be certified with a good aggregation. In
particular, if $T$ is the intersection of $S_{\lambda}$ for all good
aggregations $\lambda$, then every connected component of $T$ intersects a
component of $S$. Indeed, if $T$ had a component which did not intersect $S$,
then there would be an affine hyperplane $H$ with $H\cap T\not=\emptyset$ but
$H\cap\mathop{\rm conv}(S)=\emptyset$.

We now restrict our attention to the PDLC case. Note that here, unlike in
Sections 6 and 7, we do not assume that the spectral curve is smooth. The
following two results are key ingredients in the proof of [3, Theorem 2.17]
and our Theorem 1.4. The first states that aggregations can be improved by
translating in the direction of a positive semidefinite matrix.

######  Proposition 8.2 ([3, Proposition 9.1]).

Suppose that there is $\omega\in\mathbb{R}^{3}$ such that $Q_{\omega}\succ 0$.
Set $\Theta=\\{\theta\in\mathbb{R}^{3}\;|\;Q_{\theta}\succeq 0\\}$ and
$\Lambda_{1}=\\{\lambda\in\Lambda\;|\;\lambda\text{ is a good aggregation and
}S_{\lambda}\not=\mathbb{R}^{n}\\}$. If $\lambda\in\Lambda_{1}$ and
$\theta\in\Theta$ are such that
$\lambda^{\prime}=\lambda+\theta\in\mathbb{R}^{3}_{+}\setminus\\{0\\}$, then
$\lambda^{\prime}\in\Lambda_{1}$ and $S_{\lambda^{\prime}}\subseteq
S_{\lambda}$.

The second states that the intersection over all aggregations is given by the
intersection over aggregations with support at most two.

######  Proposition 8.3 ([3, Proposition 9.2]).

Suppose that there is $\omega\in\mathbb{R}^{3}$ such that $Q_{\omega}\succ 0$.
Set $\Lambda_{1}=\\{\lambda\in\Lambda\;|\;\lambda\text{ is a good aggregation
and }S_{\lambda}\not=\mathbb{R}^{n}\\}$ and
$\Lambda_{2}=\\{\lambda\in\Lambda_{1}\;|\;|\operatorname*{supp}(\lambda)|\leq
2\\}$. Then
$\bigcap_{\lambda\in\Lambda_{1}}S_{\lambda}=\bigcap_{\lambda\in\Lambda_{2}}S_{\lambda}$.

######  Remark 8.4.

Since the convex combination of good aggregations is a good aggregation, it
follows from Proposition 8.3 that at most six aggregations are necessary to
describe the intersection of all good aggregations when PDLC holds. This was
observed in the setting of open inequalities in [3]. We will show in Theorem
1.4 that this number can be further reduced to four in the case of closed
inequalities when $S$ has no low dimensional components and no points at
infinity.

See 3.17

###### Proof.

By Theorem 1.2, $\mathop{\overline{\rm conv}}(S)$ can be described as the
intersection of finitely many good aggregations. (Compare with [3, Theorem
2.23], as the $Q_{\lambda}\not=0$ for all nonzero
$\lambda\in\mathbb{R}^{3}_{+}$ by the hypothesis that the $Q_{i}$ are linearly
independent and that the $Q_{i}$ satisfy hidden hyperplane convexity since
they satisfy PDLC [3]). By Remark 8.4, there are
$\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(r)}$, $r\leq 6$ such that
$\mathop{\overline{\rm conv}}(S)=\bigcap_{i=1}^{6}S_{\lambda^{(i)}}$. Set
$K=\mathop{\rm cone}(\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(r)})$.

Suppose for the sake of a contradiction that $\Omega^{n}(K^{\circ})$ has two
connected components. It then follows by Lemma 5.8 that $\dim
H_{0}(X(K^{\circ},f^{h}))=2$. This implies that the set $T$ defined by the
intersection of the $S_{\lambda}$ for all good aggregations $\lambda$ has two
connected components. By Remark 8.1, this implies that $S$ intersects both
components. But then, $\mathop{\rm conv}(S)$ intersects both components, which
is a contradiction since $\mathop{\rm conv}(S)$ is connected. So, the set of
good aggregations is connected in $\Lambda$. ∎

Finally, we develop a certificate that $\mathop{\rm conv}(S)=\mathbb{R}^{n}$
under the assumption of PDLC.

######  Proposition 8.5.

If $Q_{1},Q_{2},Q_{3}$ satisfy PDLC and
$\Omega^{n}(-\mathbb{R}^{3}_{+})=\emptyset$ then $\mathop{\rm
conv}(S)=\mathbb{R}^{n}$.

###### Proof.

Suppose for the sake of a contradiction that there is
$y\in\mathbb{R}^{n}\setminus\mathop{\rm conv}(S)$. Since $\\{y\\}$ is compact,
there exist $\alpha\in\mathbb{R}^{n}$ and $\beta_{1},\beta_{2}\in\mathbb{R}$
with $\beta_{1}<\beta_{2}$ such that $\alpha^{\top}y=\beta_{2}$ and
$\alpha^{\top}x<\beta_{1}$ for all $x\in\mathop{\overline{\rm conv}}(S)$. Let
$H\subseteq\mathbb{R}^{n}$ be the hyperplane defined by
$\alpha^{\top}x=\beta_{2}$ and
$H^{h}=\\{(x,x_{n+1})\;|\;\alpha^{\top}x=\beta_{2}x_{n+1}\\}$ its
homogenization so that $\mathop{\overline{\rm conv}}(S)\cap H=\emptyset$ and
$S^{h}\cap H^{h}=\\{0\\}$. Note that
$\mathcal{V}_{\mathbb{R}}(f_{1}^{h},f_{2}^{h},f_{3}^{h})\cap H=\emptyset$ and
therefore by proposition 3.4, we have that either
$\Omega^{n}_{H}(-\mathbb{R}^{3}_{+})\not=\emptyset$ or
$H^{1}(\Omega^{n-1}_{H}(-\mathbb{R}^{3}_{+}))\not=0$. However, since the
$Q_{i}$ satisfy PDLC, so do the $Q_{i}|_{H}$ and therefore
$H^{1}(\Omega^{n-1}_{H}(-\mathbb{R}^{3}_{+}))=0$. Since
$\Omega^{n}(-\mathbb{R}^{3}_{+})=\emptyset$, it follows that
$\Omega^{n}_{H}(-\mathbb{R}^{3}_{+})=\emptyset$ by the Cauchy Interlacing
Theorem. So, no such $y$ can exist and therefore $\mathop{\overline{\rm
conv}}(S)=\mathbb{R}^{n}$. ∎

We now show that if PDLC is satisfied, $S$ has no points at infinity, and
$S=\mathrm{cl}(\mathrm{int}(S))$, then $\mathop{\overline{\rm
conv}}(S)=\bigcap_{i=1}^{r}S_{\lambda^{(i)}}$ for some $r\leq 4$. Note that
[3, Conjecture 3.2] conjectures the existence of a set defined by _strict_
quadratic inequalities which cannot be described as the intersection of fewer
than six good aggregations. It was observed in [6, Theorem 2.7] that if
$S=\mathrm{cl}(\mathrm{int}(S))$ and $\mathop{\rm
conv}(\mathrm{int}(S))=\bigcap_{\lambda\in\Lambda_{1}}\mathrm{int}(S_{\lambda})$
for some set of good aggregations $\Lambda_{1}$, then $\mathop{\overline{\rm
conv}}(S)=\bigcap_{\lambda\in\Lambda_{1}}S_{\lambda}$.

We use a similar strategy to [3] of improving aggregations in the direction of
a positive semidefinite combination. Note that if PDLC is satisfied and
$S\not=\emptyset$, then the hyperbolicity cone of $g$ which consists of
positive definite matrices does not intersect $\mathbb{R}^{3}_{+}$ away from
0. However, it is posible that the hyperbolicity cone of $g$ which contains
negative definite matrices intersects $\mathbb{R}^{3}_{+}$ and that this
hyperbolicity is contained strictly in $\mathbb{R}^{3}_{+}$.

We first deal with the case that the cone of negative definite matrices is not
strictly contained in $\mathbb{R}^{3}_{+}$ and employ the strategy of
Proposition 8.2 to improve aggregations along the direction of a positive
semidefinite matrix.

######  Proposition 8.6.

Spupose that $Q_{1},Q_{2},Q_{3}$ satisfy PDLC,
$\mathrm{int}(S)\not=\emptyset$, and $S$ has no points at infinity. Assume
that the hyperbolicity cone of $g$ which contains negative definite matrices
is not contained in $\mathbb{R}^{3}_{+}$. Then, there are
$\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(r)}$ such that $r\leq 4$ and
$\mathop{\overline{\rm conv}}(S)=\bigcap_{i=1}^{r}S_{\lambda^{(i)}}$.

###### Proof.

If the hyperbolicity cone of $g$ which contains negative definite matrices is
not contained in $\mathbb{R}^{3}_{+}$, then there is $\theta\in\mathbb{R}^{3}$
such that $Q_{\theta}\succeq 0$ and at least one component of $\theta$ is
strictly positive. Suppose without loss of generality that $\theta_{3}>0$.
Note also that at least one of $\theta_{1},\theta_{2}<0$ since otherwise
$\theta\in\mathbb{R}^{3}_{+}$ and therefore $\mathrm{int}(S)=\emptyset$.

Now, if $\lambda=\lambda_{1}e_{1}+\lambda_{2}e_{2}\in\Lambda$ is an
aggregation for $\lambda_{1},\lambda_{2}>0$, we have that
$\lambda+\epsilon\theta\in\mathbb{R}^{3}_{+}\setminus\\{0\\}$, where

$\epsilon=\min\left\\{\frac{-\lambda_{i}}{\theta_{i}}\;\middle|\;\theta_{i}<0\right\\}>0.$

So, by Proposition 8.2, $S_{\lambda+\epsilon\theta}$ is an aggregation with
$S_{\lambda+\epsilon\theta}\subseteq S_{\lambda}$ and
$\lambda+\epsilon\theta\in\mathop{\rm cone}(e_{1},e_{3})\cup\mathop{\rm
cone}(e_{2},e_{3})$. In particular, this implies that no aggregations in
$\mathop{\rm cone}(e_{1},e_{2})$ are necessary for the description of
$\mathop{\rm conv}(S)$. By Proposition 8.3 and the fact that all aggregations
in $\mathop{\rm cone}(e_{i},e_{j})$ can be described using at most two
aggregations, this in turn implies that $\mathop{\overline{\rm conv}}(S)$ can
be described as the intersection of at most four aggregations. ∎

If $n=1$ or $2$, then we are able to eliminate the possibility of six
aggregations being necessary using the structure of the curve
$\mathcal{V}_{\mathbb{R}}(g)$. As in the statement of Proposition 8.3, set

$\Lambda_{1}=\\{\lambda\in\Lambda\;|\;\lambda\text{ is a good aggregtaion and
}S_{\lambda}\neq\mathbb{R}^{n}\\},$

and
$\Lambda_{2}=\\{\lambda\in\Lambda_{1}\;|\;|\operatorname*{supp}\lambda|\leq
2\\}$. Set $\Lambda_{3}$ to be the set of unit length generators of the
extreme rays of $\Lambda_{2}$.

######  Proposition 8.7.

If $n=1$ or $n=2$, $Q_{1},Q_{2},Q_{3}$ satisfy PDLC,
$\mathrm{int}(S)\not=\emptyset$, $S=\mathrm{cl}(\mathrm{int}(S))$, $S$ has no
points at infinity, and the hyperbolicity cone of $g$ which contains negative
definite matrices is contained in $\mathbb{R}^{3}_{+}$, then there are
$\lambda^{(1)},\lambda^{(2)},\ldots,\lambda^{(r)}\in\Lambda_{2}$ such that
$r\leq 4$ and $\mathop{\overline{\rm
conv}}(S)=\bigcap_{\lambda\in\Lambda_{2}}S_{\lambda}=\bigcap_{i=1}^{r}S_{\lambda^{(i)}}$.

###### Proof.

It follows from Proposition 8.3 that $\mathop{\overline{\rm
conv}}(S)=\bigcap_{\lambda\in\Lambda_{2}}S_{\lambda}$. Since the convex
combination of good aggregations is a good aggregation if it is permissible,
it follows that $\mathop{\overline{\rm
conv}}(S)=\bigcap_{\lambda\in\Lambda_{3}}S_{\lambda}$ and therefore it
suffices to bound $|\Lambda_{3}|$. If $\lambda\in\Lambda_{3}$, then either
$\lambda$ is a standard basis vector or $g(\lambda)=0$. Note that if
$\lambda\in\Lambda_{3}$, then $Q_{\lambda}$ has exactly one negative
eigenvalue since if $Q_{\lambda}\succeq 0$ then
$\mathrm{int}(S_{\lambda})=\emptyset$, a contradiction with the hypothesis
that $\mathrm{int}(S)\not=\emptyset$ since
$\mathrm{int}(S)\subseteq\mathrm{int}(S_{\lambda})$. Note also that the set of
good aggregations is connected in $\Lambda$ by Propositions 3.15 and 3.17.

In the case $n=1$, note that for $\lambda\in\mathbb{R}^{3}_{+}$, either
$Q_{\lambda}\prec 0$ or $\lambda\in\Lambda$. Since the cone of negative
definite matrices is contained in $\mathbb{R}^{3}_{+}$, it follows that every
nonzero $\lambda\in\mathbb{R}^{3}_{+}$ with
$|\operatorname*{supp}(\lambda)|\leq 2$ has exactly one negative eigenvalue.
In particular, we have that $\Lambda$ is connected, and therefore the set of
good aggregations is also connected. It then follows that
$\Lambda_{3}=\\{e_{1},e_{2},e_{3}\\}$.

In the case $n=2$, if $\lambda\in\Lambda_{3}$ is not a standard basis vector,
then $\lambda$ lies on the non-oval component of
$\mathcal{V}_{\mathbb{R}}(g)$. Note that the line through $e_{j}$ and $e_{k}$
can only intersect this component of the spectral curve once. So, if a face
$\mathop{\rm cone}(e_{j},e_{k})$ of $\mathbb{R}^{3}_{+}$ contains two elements
of $\Lambda_{3}$, it follows that the elements are $e_{j},e_{k}$ or that
exactly one of $e_{j},e_{k}$ is an element of $\Lambda_{3}$. So, up to
reordering of the $Q_{i}$, the three possibilities for $\Lambda_{3}$ are

$\\{e_{1},e_{2},e_{3}\\},\quad\\{e_{1},\lambda^{(2)},\lambda^{(3)}\\},\text{
and}\quad\\{e_{2},e_{3},\lambda^{(2)},\lambda^{(3)}\\},$

where $\lambda^{(i)}\in\mathop{\rm cone}(e_{1},e_{i})$ is not a standard basis
vector.

In all cases, $|\Lambda_{3}|\leq 4$. ∎

We now address the case that $n\geq 3$ and the hyperbolicity cone of $g$ which
contains negative definite matrices is strictly contained in
$\mathbb{R}^{3}_{+}$. To do so, we need the following lemma which controls the
eigenvalues of matrices which are convex combinations of an element of
$\Lambda_{3}$ and a negative definite matrix.

######  Lemma 8.8.

Suppose that the hyperbolicity cone of $g$ which contains negative definite
matrices is strictly contained in $\mathbb{R}^{3}_{+}$. Suppose that
$\lambda\in\Lambda_{3}$ has $g(\lambda)=0$. Let $\omega\in\mathbb{R}^{3}$ be
such that $Q_{\omega}\prec 0$. Then, for any $t\in(0,1)$, the matrix
$Q_{t\lambda+(1-t)\omega}$ has at most $n-1$ positive eigenvalues.

###### Proof.

Consider the univariate polynomial $g(t\lambda+(1-t)\omega)$. Suppose for the
sake of a contradiction that there is $t^{*}\in(0,1)$ such that
$Q_{t^{*}\lambda+(1-t^{*})\omega}$ has $n$ positive eigenvalues. Then, since
$g(\lambda)=0$ and $Q_{\omega}$ has $n+1$ negative eigenvalues, there are