# Implicitisation and parameterisation in polynomial functors

Andreas Blatter<EMAIL_ADDRESS>Mathematical Institute, University
of Bern, Alpeneggstrasse 22, 3012 Bern, Switzerland  ,  Jan Draisma
<EMAIL_ADDRESS>Mathematical Institute, University of Bern, Sidlerstrasse
5, 3012 Bern, Switzerland; and Department of Mathematics and Computer Science,
Eindhoven University of Technology, P.O. Box 513, 5600MB Eindhoven, The
Netherlands  and  Emanuele Ventura<EMAIL_ADDRESS><EMAIL_ADDRESS>Politecnico di Torino, Dipartimento di Scienze
Matematiche “G.L. Lagrange”, Corso Duca degli Abruzzi 24 10129 Torino, Italy

###### Abstract.

In earlier work, the second author showed that a closed subset of a polynomial
functor can always be defined by finitely many polynomial equations. In
follow-up work on $\operatorname{GL}\nolimits_{\infty}$-varieties, Bik-
Draisma-Eggermont-Snowden showed, among other things, that in characteristic
zero every such closed subset is the image of a morphism whose domain is the
product of a finite-dimensional affine variety and a polynomial functor. In
this paper, we show that both results can be made algorithmic: there exists an
algorithm $\mathbf{implicitise}$ that takes as input a morphism into a
polynomial functor and outputs finitely many equations defining the closure of
the image; and an algorithm $\mathbf{parameterise}$ that takes as input a
finite set of equations defining a closed subset of a polynomial functor and
outputs a morphism whose image is that closed subset.

AB was supported by Swiss National Science Foundation grant 200021_191981

JD was partially supported by SNSF grant 200021_191981 and Vici grant
639.033.514 from the Netherlands Organisation for Scientific Research

##  1\. Introduction

###  1.1. Implicitisation

An important theme in computational algebraic geometry is implicitisation:
given a list $(\varphi_{1},\ldots,\varphi_{n})$ of polynomials in
$K[x_{1},\ldots,x_{m}]$, which represent a polynomial map $\varphi$ from the
$m$-dimensional affine space ${\mathbb{A}}^{m}$ to ${\mathbb{A}}^{n}$ over the
field $K$, the challenge is to compute equations for the Zariski closure
$\overline{\operatorname{im}(\varphi)}$ of the image of $\varphi$. This
challenge is solved, at least theoretically, by elimination using Buchberger’s
algorithm; e.g., see [CLO15, §3.3].

###  1.2. Implicitisation in families

But now consider a scenario where one is given not a single polynomial map,
but rather a family $\varphi_{i}:{\mathbb{A}}^{m_{i}}\to{\mathbb{A}}^{n_{i}}$
of polynomial maps depending on a discrete parameter $i$ that takes infinitely
many values. If the ambient spaces ${\mathbb{A}}^{m_{i}}$ and
${\mathbb{A}}^{n_{i}}$ and the polynomial map $\varphi_{i}$ vary favourably
with $i$, it is sometimes possible to find finitely many equations that
capture the image closures of all $\varphi_{i}$ at once.

###  1.3. Implicitisation over categories

To make the phrase “vary favourably” concrete, assume that $i$ ranges through
the objects of a category $C$ and ${\mathbb{A}}(i)$ is a finite-dimensional
affine space varying functorially with $i$. This means that for each
$\pi\in\operatorname{Hom}\nolimits_{C}(i,j)$ we have a linear map
${\mathbb{A}}(\pi):{\mathbb{A}}(i)\to{\mathbb{A}}(j)$ such that
${\mathbb{A}}(\operatorname{id}_{i})=\operatorname{id}_{{\mathbb{A}}(i)}$ and
${\mathbb{A}}(\sigma\circ\pi)={\mathbb{A}}(\sigma)\circ{\mathbb{A}}(\pi)$ for
$\sigma\in\operatorname{Hom}\nolimits_{C}(j,k)$. Suppose that
${\mathbb{A}}^{\prime}(i)$ is another affine space depending functorially on
$i\in C$, and finally that, for any $i\in C$, we have a polynomial map
$\varphi_{i}:{\mathbb{A}}(i)\to{\mathbb{A}}^{\prime}(i)$ such that the
following diagram commutes for every
$\pi\in\operatorname{Hom}\nolimits_{C}(i,j)$:

$\textstyle{{\mathbb{A}}(i)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{i}}$$\scriptstyle{{\mathbb{A}}(\pi)}$$\textstyle{{\mathbb{A}}^{\prime}(i)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathbb{A}}^{\prime}(\pi)}$$\textstyle{{\mathbb{A}}(j)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{j}}$$\textstyle{{\mathbb{A}}^{\prime}(j).}$

In this case, if $f$ is a polynomial equation for the image closure
$\overline{\operatorname{im}(\varphi_{j})}$ of $\varphi_{j}$, then
$f\circ{\mathbb{A}}^{\prime}(\pi)$ is a polynomial equation for
$\overline{\operatorname{im}(\varphi_{i})}$—and, since we assumed that
${\mathbb{A}}^{\prime}(\pi)$ is linear, of the same degree. We then ask:

  1. (1)

Do there exist finitely many $j\in C$ such that the equations for those
$\overline{\operatorname{im}(\varphi_{j})}$, by pulling back along the linear
maps $A^{\prime}(\pi)$ for all relevant $\pi$, define
$\overline{\operatorname{im}(\varphi_{i})}$ for all $i\in C$?

  2. (2)

If so, does there exist an algorithm for computing these finitely many $j$?

A well-known case where the answer to the two questions above is “yes” is that
where $C$ is the opposite category $\mathbf{FI^{op}}$ of the category
$\mathbf{FI}$ of finite sets with injections,
${\mathbb{A}}(i)=({\mathbb{A}}^{n})^{i}$,
${\mathbb{A}}^{\prime}(i)=({\mathbb{A}}^{n^{\prime}})^{i}$ for some fixed $n$
and $n^{\prime}$, and the maps
${\mathbb{A}}(i)\to{\mathbb{A}}(j),{\mathbb{A}}^{\prime}(i)\to{\mathbb{A}}^{\prime}(j)$
corresponding to an injection $j\to i$ are the canonical projections. In that
case, it is known that the kernel of the $\mathbf{FI}$-homomorphism
$\varphi^{*}$ dual to the $\mathbf{FI^{op}}$-polynomial map $\varphi$ is
finitely generated and can be computed using a version of Buchberger’s
algorithm, and this has been applied to problems in algebraic statistics
[Coh87, HS12, HKL13, HKL18, BD11]. The setting discussed in the current paper
is of a very different flavour in that it involves continuous symmetries
rather than discrete symmetries, and it is also significantly more
complicated. One cause for trouble is that in the setting below, we do not
actually know whether the kernel of the relevant algebra homomorphism is
finitely generated, and so we have to settle for finding set-theoretic
equations for the $\overline{\operatorname{im}(\varphi_{i})}$.

###  1.4. Implicitisation in polynomial functors

In the case where $C$ is the category $\mathbf{Vec}$ of finite-dimensional
$K$-vector spaces, and both ${\mathbb{A}}$ and ${\mathbb{A}}^{\prime}$ are
polynomial functors $\mathbf{Vec}\to\mathbf{Vec}$ (see below for definitions),
the second author established a positive answer to the first question above
[Dra19]; see Theorem 2.4.2 below.

######  Example 1.4.1.

Here are two instances of this setting:

  1. (1)

${\mathbb{A}}(V)=V^{k}$ for some positive integer $k$,
${\mathbb{A}}^{\prime}(V)=S^{3}V$, the third symmetric power of $V$, and

$\varphi_{V}(v_{1},\ldots,v_{k}):=v_{1}^{3}+\cdots+v_{k}^{3}.$

The image of $\varphi_{V}$ is the set of cubic homogeneous polynomials in
$\dim(V)$ variables of Waring rank $\leq k$, and its closure is the variety of
polynomials of border Waring rank $\leq k$.

  2. (2)

${\mathbb{A}}(V)=(S^{2}V)^{k}\times V^{k}$, ${\mathbb{A}}^{\prime}(V)=S^{3}V$,
and

$\varphi_{V}(q_{1},\ldots,q_{k},v_{1},\ldots,v_{k}):=q_{1}v_{1}+\cdots+q_{k}v_{k}.$

In this case, the image of $\varphi_{V}$ is closed [DES17] and consists of the
cubics of q-rank $\leq k$.

In the first case, it is well-known that generators of the vanishing ideal of
$\overline{\operatorname{im}(\varphi_{U})}$ for $U$ of dimension $k+1$ pull
back to generators of the ideal of $\overline{\operatorname{im}(\varphi_{V})}$
for all $V$; this is called symmetric inheritance in [LO13]. Much more is
known for small values of $k$; in particular, for $k\leq 2$, the ideal is
generated by $3\times 3$-minors of catalecticant matrices [Rai13].

In the second case, we do not know whether the ideal is finitely generated in
this sense, but [DES17] assures the existence of a number $\ell=\ell(k)$ such
that equations for $\overline{\operatorname{im}(\varphi_{U})}$ with $U$ of
dimension $\ell$ pull back to equations that define
$\overline{\operatorname{im}(\varphi_{V})}$ for all $V$ set-theoretically. No
explicit value of $\ell(k)$ is known, even for $k=2$.

In many respects, the q-rank example is much more difficult than the Waring
rank example. The main reason for this is that the source ${\mathbb{A}}$ is a
polynomial functor of degree $2$, whereas ${\mathbb{A}}$ has degree $1$ in the
first case. Nevertheless, Theorem 2.4.2 implies, for general polynomial
functors ${\mathbb{A}},{\mathbb{A}}^{\prime}$ and morphisms $\varphi$, the
existence of a $U$ such that in ${\mathbb{A}}^{\prime}(U)$ one sees enough
set-theoretic equations to define $\overline{\operatorname{im}(\varphi_{V})}$
for all $V$.

###  1.5. Main result

The main result of this paper makes Theorem 2.4.2 effective.

######  Theorem 1.5.1 (Main Theorem; see Theorem 4.1.1 for a more precise
statement).

There exists an algorithm $\mathbf{implicitise}$ that, on input polynomial
functors ${\mathbb{A}},{\mathbb{A}}^{\prime}$ and a morphism
$\varphi:{\mathbb{A}}\to{\mathbb{A}}^{\prime}$, computes a $U\in\mathbf{Vec}$
such that the equations for $\overline{\operatorname{im}(\varphi_{U})}$ pull
back to set-theoretic defining equations for
$\overline{\operatorname{im}(\varphi_{V})}$ for all $V$.

This means, in particular, that the number $\ell(k)$ for the cubics of q-rank
at most any fixed $k$ can in principle be computed.

###  1.6. Structure of the algorithm

The structure of $\mathbf{implicitise}$ is as follows. For
$U=K^{0},K^{1},K^{2},\ldots$ one computes equations for
$\overline{\operatorname{im}(\varphi_{U})}$ using Buchberger’s algorithm. By
Theorem 2.4.2 we know that at some point these equations, via pull-back,
define $\overline{\operatorname{im}(\varphi_{V})}$ for all $V$. However,
Theorem 2.4.2 does not give a criterion for when we may stop—this is different
from the algorithm in the $\mathbf{FI}$-setting: there, if one has not seen
new equations arise between the the finite set $\\{1,\ldots,n\\}$ and the
finite set $\\{1,\ldots,2n\\}$, one is guaranteed to have found a generating
set of equations [BD11].

To obtain a stopping criterion in the polynomial functor setting, we derive an
algorithm $\mathbf{parameterise}$ that, on input the polynomial equations on
${\mathbb{A}}^{\prime}(U)$ found so far, computes a parameterisation $\psi$ of
the closed subset of ${\mathbb{A}}^{\prime}$ defined by those equations. If we
can check that
$\operatorname{im}(\psi)\subseteq\overline{\operatorname{im}(\varphi)}$, then
we are done.

That such a parameterisation $\psi$ exists is guaranteed by the unirationality
theorem from [BDES21]. The algorithm $\mathbf{parameterise}$ makes this
theorem effective.

Finally, to check that $\operatorname{im}(\psi)$ lies in the closure of
$\operatorname{im}(\varphi)$, we pass to infinite dimensions and use a result
from [BDES22] that says that this happens if and only if a suitably generic
point of $\operatorname{im}(\psi_{\infty})$ can be reached as the limit of a
curve in $\operatorname{im}(\varphi_{\infty})$. In the current paper we show
that this curve, which lives in infinite-dimensional space, can be represented
in finite terms, and searched for, on a computer.

If such a curve does not exist, i.e., if $\operatorname{im}(\psi)$ is not
contained in $\overline{\operatorname{im}(\varphi)}$, then this is because the
current space $U$ is too small. In this case, the search for a curve does not
terminate. So for our algorithm $\mathbf{implicitise}$ to terminate, it is
imperative to run the search for witness curves in parallel to the search for
equations: in each step, $U$ is increased with one dimension, new equations
are computed, and a new curve search is started. We model this behaviour by
running the algorithm on countably many parallel processors. Of course,
standard results in the theory of computation imply that this algorithm can
then also run on an ordinary Turing machine.

###  1.7. Outlook

The algorithms $\mathbf{implicitise}$ and $\mathbf{parameterise}$ are very
much theoretical algorithms, and—apart from recent research on varieties over
categories—they rely on many powerful results from classical, finite-
dimensional, computational algebra: Buchberger’s algorithm, of course, but
also algorithms for computing the radical of an ideal and for computing the
minimal primes containing a radical ideal. Also, a fair amount of
representation theory of the general linear group goes into the algorithm.

At present, a general implementation of $\mathbf{implicitise}$ and
$\mathbf{parameterise}$ seems entirely out of reach. However, we believe that
our computational approach to implicitisation in polynomial functors will in
the future serve as a guide towards finding equations in concrete settings,
e.g. for the variety of cubics of q-rank $\leq 2$ from Example 1.4.1.

###  1.8. Organisation of the paper

In Section 2, we collect material that we will need in the rest of the paper:
some assumptions on the ground field, computer representations of finite-
dimensional affine varieties, polynomial functors and their closed subsets,
and morphisms between these.

In Section 3, we derive the algorithm $\mathbf{parameterise}$; see Theorem
3.1.1. In Section 4 we derive the algorithm $\mathbf{implicitise}$ and prove
our Main Theorem; see Theorem 4.1.1 for a more precise statement.

Finally, $\mathbf{implicitise}$ depends on a procedure called
$\mathbf{certify}$ that aims at certifying the existence of a curve as
discussed in §1.6. This procedure requires tools from infinite-dimensional
algebraic geometry, both from a forthcoming paper [BDES22] and from more
classical sources. In particular, it is Greenberg’s approximation theorem
(Theorem 5.5.1) from [Gre66] and its effective version in [Ron18] which allows
us to represent this curve in finite terms.

### Acknowledgments

The second author thanks Arthur Bik, Rob Eggermont, and Andrew Snowden for
numerous discussions on polynomial functors, especially on the topic of
systems of variables from §5.3 and on formal curves approximating a point in a
closure, Theorem 5.2.1.

##  2\. Preliminaries

###  2.1. The ground field

Let $K$ be a field of characteristic zero in which we can do computations on a
computer. More precisely, we want $K$ to be a computable field, and we further
require that there exists an algorithm for factoring polynomials in $K[X]$.
The class of such fields is quite large; it includes ${\mathbb{Q}}$ and its
finitely generated extensions [GP08, Appendix B].

###  2.2. Finite-dimensional affine varieties

A finite-dimensional affine variety over $K$ is represented by a finite list
of generators of a radical ideal in some polynomial ring over $K$ with
finitely many variables. A morphism between affine varieties over $K$ is
represented by a finite list of polynomials. Our algorithms will intensively
use existing algorithms for dealing with affine varieties. Good general
references are [CLO15, CLO05]. In particular, we will need algorithms for
computing the radical of an ideal; see e.g. [KL91].

If $R$ is a $K$-algebra and $h$ is an element of $R$, then we write $R[1/h]$
for the localisation. Similarly, if $B$ is a finite-dimensional affine variety
and $h$ an element of $K[B]$, then we write $B[1/h]$ for the basic open subset
defined by $h$, i.e., the affine variety with coordinate ring $K[B][1/h]$.

###  2.3. Polynomial functors

Let $\mathbf{Vec}$ be the category of finite-dimensional $K$-vector spaces. We
write $\operatorname{Hom}\nolimits(U,V)$ and $\operatorname{End}\nolimits(U)$
for the spaces of $K$-linear maps $U\to V$ and $U\to U$, respectively.

######  Definition 2.3.1.

A polynomial functor of degree at most $d$ over $K$ is a covariant functor
$P:\mathbf{Vec}\to\mathbf{Vec}$ such that for any $U,V\in\mathbf{Vec}$ the map
$P:\operatorname{Hom}\nolimits(U,V)\to\operatorname{Hom}\nolimits_{P(U),P(V)}$
is polynomial of degree at most $d$. Polynomial functors of degree at most $d$
form an abelian category in which a homomorphism $P\to Q$ is a natural
transformation, i.e., given by a linear map $\psi_{U}:P(U)\to Q(U)$ for each
$U\in\mathbf{Vec}$, such that for all
$\varphi\in\operatorname{Hom}\nolimits(U,V)$ we have
$Q(\varphi)\circ\psi_{U}=\psi_{V}\circ P(\varphi)$.

When we say polynomial functor, we will always mean a polynomial functor of
degree at most some integer. A priori, a polynomial functor seems to be given
by an infinite amount of data. But up to isomorphism, the following lemma due
to Friedlander-Suslin [FS97, Lemma 3.4] implies that it is given by a finite
amount of data only.

######  Lemma 2.3.2 ([FS97]).

The map from polynomial functors of degree at most $d$ to
$\operatorname{GL}\nolimits_{d}$-representations that assigns to $P$ the
vector space $P(K^{d})$ with the algebraic group homomorphism
$\operatorname{GL}\nolimits_{d}\to\operatorname{GL}\nolimits(P(K^{d})),\varphi\mapsto
P(\varphi)$ is an equivalence of abelian categories from polynomial functors
of degree at most $d$ to polynomial
$\operatorname{GL}\nolimits_{d}$-representations of degree at most $d$.

Here, a homomorphism
$\operatorname{GL}\nolimits_{n}\to\operatorname{GL}\nolimits(V)$ of algebraic
groups is called polynomial of degree at most $d$ if it extends to a
polynomial map
$\operatorname{End}\nolimits(K^{n})\to\operatorname{End}\nolimits(V)$ of
degree at most $d$.

The Friedlander-Suslin lemma is relevant to us for two reasons. First, since
polynomial representations of $\operatorname{GL}\nolimits_{d}$ are completely
reducible and the irreducible ones are completely classified by combinatorial
data, the same holds for polynomial functors. The consequence is that each
polynomial functor of degree at most $d$ is a direct sum of Schur functors:
functors of the form
$S_{\lambda}:V\mapsto\operatorname{Hom}\nolimits(U_{\lambda},V^{\otimes e})$,
where $e\leq d$, $\lambda$ is a partition of $e$, and $U_{\lambda}$ is the
corresponding irreducible representation of the symmetric group on $e$
letters. So we may represent a polynomial functor by a finite tuple of
partitions. Second, the proof of the Friedlander-Suslin lemma is completely
constructive: it can be transformed into algorithms that compute, from a
polynomial representation $\rho$ of $\operatorname{GL}\nolimits_{d}$ of degree
$\leq d$ corresponding to a polynomial functor $P$:

  * •

(a basis for) $P(V)$ on input $V$;

  * •

(a matrix for) $P(\varphi):P(U)\to P(V)$ on input a linear map $\varphi:U\to
V$; and

  * •

(a matrix for) $\psi_{V}:P(V)\to Q(V)$ on input a second polynomial
$\operatorname{GL}\nolimits_{d}$-representation of degree $\leq d$
representing the polynomial functor $Q$, as well as homomorphism of
$\operatorname{GL}\nolimits_{d}$-representations.

We will not make these algorithms explicit here.

Each polynomial functor $P$ of degree at most $d$ is a direct sum of
homogeneous polynomial functors: $P=P_{0}\oplus\cdots\oplus P_{d}$, where

$P_{e}(V):=\\{p\in P(V)\mid P(t\operatorname{id}_{V})p=t^{e}p\\}.$

In particular, $P_{0}$ is a degree-$0$ polynomial functor, which assigns a
fixed vector space, also denoted $P_{0}$, to every $V$ and the identity
$\operatorname{id}_{P_{0}}$ to every linear map
$\varphi\in\operatorname{Hom}\nolimits(U,V)$. We call $P$ pure if $P_{0}=0$,
and we call $P_{1}\oplus\cdots\oplus P_{d}$ the pure part of $P$.

###  2.4. Closed subsets of polynomial functors

######  Definition 2.4.1.

A closed subset $X$ of a polynomial functor $P$ over $K$ is the data of a
closed subvariety $X(V)$ of $\overline{K}\otimes P(V)$ defined over $K$ such
that for each $\varphi\in\operatorname{Hom}\nolimits(U,V)$ the linear map
$1\otimes P(\varphi)$ maps $X(U)$ into $X(V)$.

We take points with coordinates in $\overline{K}$ because $K$ might not be
large enough to see all points. On the other hand, our algorithm will always
work over $K$ itself, and all varieties will be defined over $K$. In fact, we
shall usually drop the $\overline{K}$ from the notation and just write closed
subvariety $X(V)$ of $P(V)$ and $P(\varphi):X(U)\to X(V)$ in the above
setting. This is not different from the classical setting of implicitisation,
where one computes the equations of the Zariski closure in $\overline{K}^{n}$
of the image of a polynomial map $\overline{K}^{m}\to\overline{K}^{n}$ which
is defined over $K$.

For a closed subset $X$ of a polynomial functor $P$ and a linear map
$\varphi\in\operatorname{Hom}\nolimits(U,V)$, we will write
$X(\varphi):X(U)\to X(V)$ for the restriction of $P(\varphi)$ to $X(U)$. Note
that, in particular, the map $\operatorname{GL}\nolimits(V)\times X(V)\to
X(V),(g,x)\mapsto X(g)(x)$ defines an algebraic action of
$\operatorname{GL}\nolimits(V)$ on $X(V)$, for each $V\in\mathbf{Vec}$. So
this paper is concerned with (typically) highly symmetric varieties, related
by linear maps coming from linear maps between distinct vector spaces.

If $X$ is a closed subset of $P$, then the variety $B:=X(0)$ is a closed
subvariety of $P(0)=P_{0}$. For each $V\in\mathbf{Vec}$, the zero map
$0_{V,0}:V\to 0$ maps $X(V)$ to $X(0)$, and indeed onto $X(0)$, because for
any $p_{0}\in X(0)\subseteq P(0)$ we have

$p_{0}=P(0_{V,0}\circ 0_{0,V})p_{0}=P(0_{V,0})(P(0_{0,V})p_{0})\in
P(0_{V,0})(X(V))$

where we have used that $P$ is a functor and $X$ is a closed subset of $P$.
Hence for each $V$, $X(V)$ is a closed subset of $B\times P^{\prime}(V)$,
where $P^{\prime}$ is the pure part of $P$, such that $X(V)$ maps surjectively
to $B$. We will also say that $X$ is a closed subset of $B\times P^{\prime}$.

The following theorem with its corollary implies that a closed subset $X$ of a
polynomial functor $P$ can be represented on a computer.

######  Theorem 2.4.2 ([Dra19]).

Let $P$ be a polynomial functor and let $X_{1}\supseteq X_{2}\supseteq\ldots$
be a descending chain of closed subsets of $P$. Then this chain stabilizes,
i.e., there exists $n_{0}$ such that $X_{n_{0}}=X_{n_{0}+1}=\ldots$

######  Corollary 2.4.3.

For each closed subset $X\subseteq P$ there exists a vector space
$U\in\mathbf{Vec}$ with the property that for all $V\in\mathbf{Vec}$ we have

$X(V)=\bigcap_{\varphi\in\operatorname{Hom}\nolimits(V,U)}P(\varphi)^{-1}(X(U)).$

###### Proof.

For every $n\in{\mathbb{N}}$ consider the closed subset $X_{n}$ in $P$ defined
by

$X_{n}(V):=\bigcap_{\varphi\in\operatorname{Hom}\nolimits(V,K^{n})}P(\varphi)^{-1}(X(K^{n})).$

Using that $P$ is a polynomial functor and $X$ is a closed subset, we can see
that for all $n$, $X_{n}(K^{n})=X(K^{n})$ and that $X_{1}\supseteq
X_{2}\supseteq\ldots$ is a descending chain. By Theorem 2.4.2 this chain
stabilizes at, say, $X_{n_{0}}$, and hence for $n\geq n_{0}$,
$X(K^{n})=X_{n}(K^{n})=X_{n_{0}}(K^{n})$. Using that $X$ is a closed subset,
we get the same for $n<n_{0}$. Hence, $X=X_{n_{0}}$, and the corollary holds
for $U=K^{n_{0}}$. ∎

Consequently, if $f_{1},\ldots,f_{k}\in K[P(U)]$ are defining equations for
$X(U)$, then the pull-backs of the $f_{i}$ along all linear maps
$P(\varphi):P(V)\to P(U)$ cut out $X(V)$. We then write
$X=\mathcal{V}_{P}(f_{1},\ldots,f_{k})$. The tuple consisting of $P$, $U$, and
$(f_{1},\ldots,f_{k})$ together form a computer representation of $X$.
Slightly more generally, we will often represent $X$ as follows.

######  Definition 2.4.4.

Let $B$ be a finite-dimensional affine variety, $P$ a pure polynomial functor,
$U$ a finite-dimensional vector space, and $f_{1},\ldots,f_{k}\in K[B\times
P(U)]$. The tuple $(B,P,U,f_{1},\ldots,f_{k})$ is called the implicit
representation of the closed subset $X$ of $B\times P$ defined by

$X(V)=\\{(b,p^{\prime})\mid\forall\varphi\in\operatorname{Hom}\nolimits(V,U)\
\forall i=1,\ldots,r:f_{i}(b,P(\varphi)p^{\prime})=0\\}.$

Again by Corollary 2.4.3, every closed subset of $B\times P$ admits an
implicit representation. Here we allow that some of the $f_{i}$ are nonzero
elements of $K[B]$, so that $X$ does not map surjectively to $B$ but rather
onto a closed subvariety of $B$.

###  2.5. Irreducibility

######  Definition 2.5.1.

Let $B$ be a finite-dimensional affine variety, $Q$ a pure polynomial functor,
and $X$ a closed subset of $B\times Q$. Then $X$ is called irreducible if
$X\neq\emptyset$ and whenever $X=X_{1}\cup X_{2}$ where $X_{1},X_{2}$ are
closed subsets of $B\times Q$, we have $X=X_{1}$ or $X=X_{2}$.

A straightforward check shows that $X$ is irreducible if and only if $X(V)$ is
irreducible for each $V\in\mathbf{Vec}$; in particular, $B\times Q$ is
irreducible if and only if $B$ is irreducible. Note that since $K$ may not be
algebraically closed, irreducibility in our sense may not imply irreducibility
over $\overline{K}$.

###  2.6. Gradings and ideals

For any polynomial functor $P=P_{0}\oplus\cdots\oplus P_{d}$, and any
$V\in\mathbf{Vec}$, the coordinate ring
$K[P(V)]\cong\bigotimes_{e=0}^{d}K[P_{e}(V)]$ is a graded polynomial ring in
which the coordinates on $K[P_{e}(V)]$ are given the degree $e$. A polynomial
$f$ is homogeneous of degree $n$ with respect to this grading if and only if
$f(P(t\cdot\operatorname{id}_{V})p)=t^{n}f(p)$ for all $p\in P(V)$. If $X$ is
a closed subset of $P$, then $X(V)$ is preserved under
$P(t\cdot\operatorname{id}_{V})$ for all $t$, and hence the ideal of $X(V)$ is
homogeneous.

Similarly, for $B$ a finite-dimensional affine variety and $P$ a pure
polynomial functor, $K[B\times P(V)]$ has a standard grading in which the
elements of $K[B]$ have degree $0$, and the ideal of any closed subset $X$ of
$B\times P$ is homogeneous.

We find that the coordinate rings $K[X(V)]$, for $X$ a closed subset of a
polynomial functor $P$ of degree at most $d$ (or for $X$ a closed subset of
$B\times P$ with $P$ a pure polynomial functor of degree $d$), have standard
gradings and are generated in degree at most $d$. A straightforward
computation shows that for each $\varphi\in\operatorname{Hom}\nolimits(U,V)$,
the pullback $X(\varphi)^{*}:K[X(V)]\to K[X(U)]$ is a graded $K$-algebra
homomorphism.

###  2.7. Morphisms

######  Definition 2.7.1.

Let $X,Y$ be closed subsets of polynomial functors. Then a morphism
$\alpha:X\to Y$ is given by a morphism $\alpha_{V}:X(V)\to Y(V)$ of affine
varieties over $K$ for each $V\in\mathbf{Vec}$ such that for all
$\varphi\in\operatorname{Hom}\nolimits(U,V)$ we have
$Y(\varphi)\circ\alpha_{U}=\alpha_{V}\circ X(\varphi)$.

By taking for $\varphi$ scalar multiples of the identity, one finds that the
pull-back $\alpha_{V}^{*}$ is a graded $K$-algebra homomorphism.

Lemma 2.7.3 ensures that we can represent morphisms on a computer. First, the
following generalises a well-known property of finite-dimensional affine
varieties (see [Bik20, Proposition 1.3.22]).

######  Lemma 2.7.2.

If $X$ is a closed subset of a polynomial functor $P$ and $Y$ is a closed
subset of a polynomial functor $Q$, then any morphism $X\to Y$ extends to a
morphism $P\to Q$.

######  Lemma 2.7.3.

Let $X,Y$ be closed subsets of polynomial functors of degree at most $d$. Then
a morphism $\alpha:X\to Y$ is uniquely determined by
$\alpha_{K^{d}}:X(K^{d})\to Y(K^{d})$, and this unique determination is
algorithmic in the sense that if $\alpha_{K^{d}}$ is known, then $\alpha_{V}$
can be computed for any $V\in\mathbf{Vec}$.

###### Proof.

By Lemma 2.7.2, $\alpha$ extends to a morphism $\beta:P\to Q$, where $X$,$Y$
are closed subsets in the degree-$\leq d$ polynomial functors $P,Q$. Now for
each $e=0,\ldots,d$, the restriction of $\beta_{V}^{*}:Q_{e}(V)^{*}\to
K[P(V)]_{e}$ defines, as $V$ varies, a homomorphism from the polynomial
functor $V^{*}\mapsto Q_{e}(V)^{*}$ to the polynomial functor $V^{*}\mapsto
K[P(V)]_{e}$, both of degree $e$ (except that $K[P(V)]_{e}$ is not finite-
dimensional if $P_{0}\neq 0$, but we may replace it by its image). Then we
apply the Friedlander-Suslin lemma to conclude that this homomorphism is
uniquely determined by its evaluation at $V=K^{d}$. The algorithmicity follows
from the algorithmicity of the Friedlander-Suslin lemma. ∎

######  Remark 2.7.4.

A special instance of the lemma is when $X=P$ and $Y=Q$, with $P,Q$ pure
polynomial functors of degree $\leq d$. In this case, the space of morphisms
$\mathbf{Map}(P,Q)$ is a finite-dimensional vector space over $K$, namely, the
direct sum for $e=1,\ldots,d$ of the space of
$\operatorname{GL}\nolimits_{d}$-equivariant linear maps $Q_{e}(V)^{*}\to
K[P(V)]_{e}$, where $V=K^{d}$. This space will be important to us towards the
end of the paper.

######  Remark 2.7.5.

We will often consider morphisms $\alpha:A\times P\to B\times Q$, where $A,B$
are finite-dimensional varieties over $K$ and $P,Q$ are pure polynomial
functors. Such a morphism decomposes into a morphism $\alpha^{(0)}:A\to B$ and
a morphism $\alpha^{(1)}:A\times P\to Q$; here we use that $\alpha_{V}^{*}$
preserves the degree and that the coordinates on $B$ have degree zero (see
§2.6), and hence their images cannot involve the positive-degree coordinates
on $P$. If $A$ is irreducible, then $\alpha^{(1)}$ can be thought of as a
$K(A)$-valued point of the finite-dimensional affine space
$\mathbf{Map}(P,Q)$; this will be a useful point of view later.

###  2.8. Shifting

######  Definition 2.8.1.

Any fixed $U\in\mathbf{Vec}$ defines a covariant polynomial functor
$\mathrm{Sh}_{U}:\mathbf{Vec}\to\mathbf{Vec}$ of degree $1$ by
$\mathrm{Sh}_{U}(V)=U\oplus V$ and, for $\varphi:V\to W$,
$\mathrm{Sh}_{U}(\varphi)=\operatorname{id}_{U}\oplus\varphi$. If $P$ is a
polynomial functor of degree $d$, then
$\mathrm{Sh}_{U}P:=P\circ\mathrm{Sh}_{U}$ is also a polynomial functor of
degree $d$, called the shift over $U$ of $P$.

The top-degree parts of $\mathrm{Sh}_{U}P$ and $P$ are canonically isomorphic
[Dra19, Lemma 14]. If $X$ is a closed subset of $P$, then
$\mathrm{Sh}_{U}X:=X\circ\mathrm{Sh}_{U}$ is a closed subset of
$\mathrm{Sh}_{U}P$. Note that $(\mathrm{Sh}_{U}X)(0)=X(U)$, so shifting has
the effect of making the finite-dimensional base variety larger. More
precisely, if $X$ is a closed subset of $B\times P$ with $P$ a pure polynomial
functor, then $\mathrm{Sh}_{U}X$ is a closed subset of $(B\times P(U))\times
P^{\prime}$ where $P^{\prime}$ is the pure part of $\mathrm{Sh}_{U}P$.

###  2.9. An order on polynomial functors

######  Definition 2.9.1.

We call a polynomial functor $Q$ smaller than a polynomial functor $P$ if the
two are not isomorphic and for the largest $e$ such that $Q_{e}$ is not
isomorphic to $P_{e}$, the former is a quotient of the latter.

Using the Friedlander-Suslin lemma, one can show that this is a well-founded
order on polynomial functors [Dra19, Lemma 12].

###  2.10. Summary

We have now indicated computer representations for all the mathematical
objects that we will need below: finite-dimensional affine varieties,
polynomial functors, closed subsets of the latter and morphisms between these.
Furthermore, we have introduced two tools in the design and analysis of our
algorithms: shifting and a well-founded order on polynomial functors.

##  3\. Parameterisation

###  3.1. The result

The goal of this section is to prove the following theorem.

######  Theorem 3.1.1.

There exists an algorithm $\mathbf{parameterise}$ that, on input a finite-
dimensional affine variety $B$, a pure polynomial functor $Q$, a finite-
dimensional vector space $U$ over $K$, and elements $f_{i}\in K[B\times Q(U)]$
for $i=1,\ldots,k$, computes $(A;P;\beta)$ where $A$ is a finite-dimensional
affine variety, $P$ is a pure polynomial functor, and $\beta$ is a morphism
$A\times P\to B\times Q$, defined over $K$, such that for each finite-
dimensional $K$-vector space $V$ we have

$\displaystyle\beta(A(\overline{K})\times\overline{K}\otimes P(V))$
$\displaystyle=\\{(b,q)\in B(\overline{K})\times\overline{K}\otimes
Q(V)\mid\forall i\ \forall\varphi\in\operatorname{Hom}\nolimits_{K}(V,U):\
f_{i}(b,1\otimes Q(\varphi)q)=0\\}.$

Here, $Z(\overline{K})$ means the $\overline{K}$-points of a finite-
dimensional affine variety $Z$ defined over $K$, i.e., the set of $K$-algebra
homomorphisms $K[Z]\to\overline{K}$. If $Z$ is just a vector space over $K$,
then this means $\overline{K}\otimes Z$. In what follows, to simplify
notation, we will supress $\overline{K}$ and write $(b,q)\in B\times Q(V)$
when we really mean $(b,q)\in B(\overline{K})\times Q(V)(\overline{K})$, and
for a $\varphi\in\operatorname{Hom}\nolimits_{K}(V,W)$ we will write
$(b,Q(\varphi)q)\in B\times Q(W)$ instead of $(b,(1\otimes Q(\varphi))q)\in
B(\overline{K})\times Q(W)(\overline{K})$. Similarly, we will write $X(V)$
even when we mean $X(V)(\overline{K})$.

We write $X:=\mathcal{V}_{B\times Q}(f_{1},\ldots,f_{k})$, so that $X(V)$ is
the set on the right-hand side above. This is the closed subset of $B\times Q$
that we want to parameterise.

######  Remark 3.1.2.

We allow $A$ to be reducible, even when $B$ is irreducible. The fact that
$\alpha:A\times P\to B\times Q$ as in the theorem exists is [Bik20, Theorem
4.2.5]; see also [BDES21, Proposition 5.6]. If $B$ is indeed irreducible, then
for some irreducible component $A^{\prime}$ of $A$ the restriction of $\alpha$
to $A^{\prime}\times P$ is dominant into $\mathcal{V}_{B\times
Q}(f_{1},\ldots,f_{k})$.

###  3.2. Smearing out equations

Before proving the theorem, we introduce an algorithm that computes the
equations for any single instance $X(V)$.

######  Proposition 3.2.1.

There exists an algorithm $\mathbf{smear}$, that, on the same input as
$\mathbf{parameterise}$ plus a finite-dimensional vector space $V$, outputs
generators of the radical ideal of $X(V)\subseteq K[B\times Q(V)]$.

###### Proof.

The algorithm $\mathbf{smear}(B;Q;U;f_{1},\ldots,f_{k};V)$ proceeds as
follows: choose identifications $U=K^{m}$ and $V=K^{n}$ and construct the
generic matrix

$\psi:=\sum_{i=1}^{\dim U}\sum_{j=1}^{\dim V}z_{ij}E_{ij}\in
K[(z_{ij})_{ij}]\otimes\operatorname{Hom}\nolimits_{K}(V,U),$

where the $E_{ij}$ form the standard basis of
$\operatorname{Hom}\nolimits_{K}(V,U)$. Then compute $Q(\psi)y\in
K[(z_{ij})_{ij}]\otimes Q(U)$, where $y=(y_{1},\ldots,y_{\dim Q(V)})^{T}$
represents a point of $Q(V)\cong K^{\dim Q(V)}$ with variables as coordinates,
substitute $Q(\psi)y$ into the $f_{i}$, and expand as a polynomial in the
$z_{ij}$ with coefficients in $K[B][y_{1},\ldots,y_{\dim Q(V)}]\cong K[B\times
Q(V)]$. Finally, return generators of the radical of the ideal generated by
all these coefficients. The correctness of this algorithm follows from the
fact that those coefficients span the same space as the images of the $f_{i}$
under pull-back along the linear maps $Q(\varphi):Q(V)\to Q(U)$ for all
$\varphi\in\operatorname{Hom}\nolimits(V,U)$. ∎

###  3.3. The parameterisation algorithm

The algorithm $\mathbf{parameterise}$ is recursive and proceeds as follows;
the algorithmic part is written in normal font, text that will be used in the
analysis in italic. The proofs of termination and correctness are below.

  1. (1)

If $Q=0$, then compute the variety $A\subseteq B$ defined by
$f_{1},\ldots,f_{k}$ via a radical ideal computation, return
$(A,0,A\hookrightarrow B)$, and exit.

  2. (2)

Decompose $Q=Q^{\prime}\oplus R$ where $R$ is an irreducible subfunctor of the
top-degree part of $Q$.

This corresponds to choosing a partition in the tuple representing $Q$. Let
$x_{1},\ldots,x_{n}$ be a basis of $R(U)^{*}$. We regard elements in
$K[B\times Q(U)]\cong K[B\times Q^{\prime}(U)]\otimes K[R(U)]\cong K[B\times
Q^{\prime}(U)][x_{1},\ldots,x_{n}]$ as polynomials in $x_{1},\ldots,x_{n}$
with coefficients in $K[B\times Q^{\prime}(U)]$.

  3. (3)

Compute

$(f_{1}^{\prime},\ldots,f^{\prime}_{r}):=\mathbf{smear}(B;Q;U;f_{1},\ldots,f_{k};U)$

and a Gröbner basis $(g_{1},\ldots,g_{l})$ of the eliminiation ideal in
$K[B\times Q^{\prime}(U)]$ obtained by eliminating all $x_{i}$ from
$f_{1}^{\prime},\ldots,f^{\prime}_{r}$.

The elements $f_{1}^{\prime},\ldots,f^{\prime}_{r}$ generate the radical ideal
of $X(U)$ in $B\times(Q^{\prime}(U)\oplus R(U))$. Let $X^{\prime}(V)$ be the
closure of the image of $X(V)$ under projection $B\times Q(V)\to B\times
Q^{\prime}(V)$, so $X^{\prime}$ is a closed subset of $B\times Q^{\prime}$.
Then $g_{1},\ldots,g_{l}$ generate the radical ideal of $X^{\prime}(U)$ by
construction; and, as we will see below in the proof of correctness, we have
$X^{\prime}=\mathcal{V}_{B\times Q^{\prime}}(g_{1},\ldots,g_{l})$.

  4. (4)

In each $f_{i}$, replace each coefficient by its normal form modulo
$g_{1},\ldots,g_{l}$.

This has the effect that all coefficients that vanish identically on
$X^{\prime}(U)$ are set to zero.

  5. (5)

If all $f_{i}$ are now zero, then compute

$(A^{\prime};P^{\prime};\alpha^{\prime}):=\mathbf{parameterise}(B;Q^{\prime};U;g_{1},\ldots,g_{l}),$

output $(A^{\prime},P^{\prime}\oplus
R,\alpha^{\prime}\times\operatorname{id}_{R})$, and exit.

  6. (6)

Pick the minimal $i$ for which $f_{i}$ is nonzero and let $x_{j}$ be a
variable that appears in some monomial in $f_{i}$ with a nonzero coefficient.

There cannot be degree-$0$ elements among the $f_{i}$, because these would lie
in the elimination ideal and hence have been reduced to zero.

  7. (7)

Compute the partial derivative $h:=\partial f_{i}/\partial x_{j}\in K[B\times
Q^{\prime}(U)][x_{1},\ldots,x_{n}]$.

By construction, this $h$ is nonzero and its coefficients, which are a subset
of the coefficients of $f_{i}$ up to some positive integer scalars, do not lie
in the ideal generated by $g_{1},\ldots,g_{l}$.

  8. (8)

Compute

$(A^{\prime};P^{\prime};\alpha^{\prime}):=\mathbf{parameterise}(B;Q;U;h,f_{1},\ldots,f_{k}).$

  9. (9)

Compute $\tilde{Q}:=(\mathrm{Sh}_{U}Q)/R$ via [FH91, Exercise 6.11], determine
the pure part $Q^{\prime\prime}$ of $\tilde{Q}$, and compute
$B^{\prime\prime}:=(B\times Q(U))[1/h]$.

So we have
$(B\times\mathrm{Sh}_{U}Q)[1/h]=B^{\prime\prime}\times(Q^{\prime\prime}\oplus
R)$.

  10. (10)

Compute

$(f_{1}^{\prime\prime},\ldots,f_{s}^{\prime\prime}):=\mathbf{smear}(B;Q;U;f_{1},\ldots,f_{k};U\oplus
U)$

and replace each $f_{i}^{\prime\prime}$ by its image in
$K[B^{\prime\prime}\times(Q^{\prime\prime}(U)\oplus R(U))]$ under the map
$K[B\times Q(U\oplus U)]\to K[B^{\prime\prime}\times(Q^{\prime\prime}(U)\oplus
R(U))]$ dual to the inclusion

$\displaystyle B^{\prime\prime}\times(Q^{\prime\prime}(U)\oplus R(U))$
$\displaystyle\subseteq B\times Q(U)\times(Q^{\prime\prime}(U)\oplus R(U))$
$\displaystyle=B\times(\mathrm{Sh}_{U}Q)(U)=B\times Q(U\oplus U).$

The elements $f_{1}^{\prime\prime},\ldots,f_{s}^{\prime\prime}$ generate the
radical ideal of $(\mathrm{Sh}_{U}X)(U)[1/h]$ in
$B^{\prime\prime}\times(Q^{\prime\prime}(U)\oplus R(U))$. As we will see in
the proof of correctness, we have
$(\mathrm{Sh}_{U}X)[1/h]=\mathcal{V}_{B^{\prime\prime}\times(Q^{\prime\prime}\oplus
R)}(f_{1}^{\prime\prime},\ldots,f_{s}^{\prime\prime})$.

  11. (11)

Using Buchberger’s algorithm to eliminate the coordinates on $R(U)$ from
$f_{1}^{\prime\prime},\ldots,f_{s}^{\prime\prime}$, compute equations
$g^{\prime\prime}_{1},\ldots,g^{\prime\prime}_{t}$ for the projection of the
finite-dimensional affine variety $(\mathrm{Sh}_{U}X)(U)[1/h]$ into
$B^{\prime\prime}\times Q^{\prime\prime}(U)$.

Recall from [Dra19, Lemma 25] that the projection
$(B^{\prime\prime}\times(Q^{\prime\prime}\oplus R))\to B^{\prime\prime}\times
Q^{\prime\prime}$ restricts to a closed embedding from
$(\mathrm{Sh}_{U}X)[1/h]$ to a closed subset $X^{\prime\prime}$ of the latter
space. We will see below that
$X^{\prime\prime}=\mathcal{V}_{B^{\prime\prime}\times
Q^{\prime\prime}}(g^{\prime\prime}_{1},\ldots,g^{\prime\prime}_{t})$.

  12. (12)

Compute the inverse $\iota:X^{\prime\prime}\to(\mathrm{Sh}_{U}X)[1/h]$.

By Lemma 2.7.3, this inverse is uniquely determined by its instance
$\iota_{V}$ with $\dim V=\deg(Q)$.

  13. (13)

Compute

$(A^{\prime\prime};P^{\prime\prime};\alpha^{\prime\prime}):=\mathbf{parameterise}(B^{\prime\prime};Q^{\prime\prime};U;g^{\prime\prime}_{1},\ldots,g^{\prime\prime}_{t}),$

output

$(A^{\prime}\sqcup A^{\prime\prime},P^{\prime}\oplus
P^{\prime\prime},\alpha^{\prime}\sqcup(\pi\circ\iota\circ\alpha^{\prime\prime})),$

where $\pi:B\times(\mathrm{Sh}_{U}Q)\to B\times Q$, evaluated at $V$, equals
$\operatorname{id}_{B}\times Q(0_{U,0}\oplus\operatorname{id}_{V})$, and exit.

Here $\alpha^{\prime}$ is regarded as a map $A^{\prime}\times(P^{\prime}\oplus
P^{\prime\prime})\to B\times Q$ that ignores the argument from
$P^{\prime\prime}$, and similarly $\alpha^{\prime\prime}$ ignores the
component in $P^{\prime}$.

###  3.4. Termination of $\mathbf{parameterise}$

###### Proof of termination of $\mathbf{parameterise}$..

If, on some input, $\mathbf{parameterise}$ would not terminate, then this were
due to an infinite chain of recursive calls to itself. In the recursive calls
in steps (5) and (13), the polynomial functors $Q^{\prime}$ and
$Q^{\prime\prime}$, respectively, are smaller than $Q$ in the order from §2.9,
whereas $Q$ remains the same in the call in step (8). Since the order on
polynomial functors is well-founded, apart from a finite initial segment, the
infinite chain consists entirely of consecutive calls in step (8). Now note
that, after each such call, $X^{\prime}(U)$ either remains constant or becomes
smaller. As long as it remains constant, i.e., the list $(g_{1},\ldots,g_{l})$
remains constant, the degree in $x_{1},\ldots,x_{n}$ of the first equation
keeps dropping. Hence after finitely many such steps, $X^{\prime}(U)$ becomes
strictly smaller. It follows that $X^{\prime}(U)$ becomes smaller infinitely
often, but this contradicts the Noetherianity of the finite-dimensional
variety $B\times Q^{\prime}(U)$. ∎

###  3.5. Correctness of $\mathbf{parameterise}$

###### Proof that $\mathbf{parameterise}$ returns the correct output..

We now prove that the output of the algorithm is correct. This is immediate if
the algorithm exits in step (1).

If it exits in step (5), then by Lemma 3.5.1 below, $X=X^{\prime}\times R$,
which is parameterised by $\alpha^{\prime}\times\operatorname{id}_{R}$ for a
parameterisation $\alpha^{\prime}$ of $X^{\prime}$. Moreover, by the same
lemma, $X^{\prime}$ is defined by $g_{1},\ldots,g_{l}$.

Finally, assume that the algorithm exits in step (13). We need to show that
the union of the images of $\alpha^{\prime}$ and
$\pi\circ\iota\circ\alpha^{\prime\prime}$ (on $\overline{K}$-points) equals
$X$.

First consider $\alpha^{\prime}$, which is computed in step (8). The closed
subset $Y:=\mathcal{V}_{B\times Q}(f_{1},\ldots,f_{k},h)$ parameterised by
$\alpha^{\prime}$ is clearly contained in $X$, so
$\alpha^{\prime}:A^{\prime}\times P^{\prime}\to B\times Q$ has image on
$\overline{K}$-points contained in the $\overline{K}$-points of $X$.

Next we argue that $\mathcal{V}_{B^{\prime\prime}\times(Q^{\prime\prime}\oplus
R)}(f_{1}^{\prime\prime},\ldots,f_{s}^{\prime\prime})$ is precisely
$(\mathrm{Sh}_{U}X)[1/h]$. First, by construction,
$f_{1}^{\prime\prime},\ldots,f_{s}^{\prime\prime}$ generate the ideal of
$(\mathrm{Sh}_{U}X)[1/h](U)$ in $(B\times Q(U\oplus U))[1/h]$; this shows the
containment $\supseteq$. For the opposite containment we argue that if $V$ is
any finite-dimensional $K$-vector space and $(b,q)\in B\times Q(U\oplus V)$
has the property that for all $\varphi\in\operatorname{Hom}\nolimits(V,U)$ the
point $(b,Q(\operatorname{id}_{U}\oplus\varphi)q)\in B\times Q(U\oplus U)$
lies in $X(U\oplus U)$, then $(b,q)\in X(U\oplus V)$. To show that $(b,q)\in
X(U\oplus V)$, since $X$ is defined by its equations in $K[B\times Q(U)]$, it
suffices to prove that for each $\psi\in\operatorname{Hom}\nolimits(U\oplus
V,U)$ we have $(b,Q(\psi)q)\in X(U)$. Let
$\varphi:=\psi|_{V}\in\operatorname{Hom}\nolimits(V,U)$ be the restriction of
$\psi$ to $V$. Then

$\ker(\operatorname{id}_{U}\oplus\varphi:U\oplus V\to U\oplus
U)=\ker(\varphi)\subseteq\ker(\psi)$

and hence the linear map $\psi$ factors as
$\psi^{\prime}\circ(\operatorname{id}_{U}\oplus\varphi)$ for some
$\psi^{\prime}\in\operatorname{Hom}\nolimits(U\oplus U,U)$. Hence, since $Q$
is a functor,

$(b,Q(\psi)q)=(b,Q(\psi^{\prime})(Q(\operatorname{id}_{U}\oplus\varphi)q))\in
X(U)$

because $(b,Q(\operatorname{id}_{U}\oplus\varphi)q)\in X(U\oplus U)$ and
$\operatorname{id}_{B}\times Q(\psi^{\prime})$ maps $X(U\oplus U)$ into
$X(U)$. This concludes the proof that
$\mathcal{V}_{B^{\prime\prime}\times(Q^{\prime\prime}\oplus
R)}(f_{1}^{\prime\prime},\ldots,f_{s}^{\prime\prime})=(\mathrm{Sh}_{U}X)[1/h]$.

Next, by [Dra19, Lemma 25], the projection
$B^{\prime\prime}\times(Q^{\prime\prime}\oplus R)\to B^{\prime\prime}\times
Q^{\prime\prime}$ restricts to an isomorphism embedding from
$(\mathrm{Sh}_{U}X)[1/h]$ to a closed subset of $B^{\prime\prime}\times
Q^{\prime\prime}$, and by Lemma 3.5.2 below, the equations of
$X^{\prime\prime}$ can be pulled back from the equations of
$X^{\prime\prime}(U)$, so we have indeed
$X^{\prime\prime}=\mathcal{V}_{B^{\prime\prime}\times
Q^{\prime\prime}}(g^{\prime\prime}_{1},\ldots,g^{\prime\prime}_{t})$.

Finally, consider step 13. A straightforward calculation shows that setting
$\pi_{V}:=\operatorname{id}_{B}\times Q(0_{U,0}\oplus\operatorname{id}_{V})$
does indeed yield a morphism $B\times\mathrm{Sh}_{U}Q\to B\times Q$ that maps
$\mathrm{Sh}_{U}X$ into $X$. We need to show that if
$\alpha^{\prime\prime}:A^{\prime\prime}\times P^{\prime\prime}\to
B^{\prime\prime}\times Q^{\prime\prime}$ is a morphism parameterising
$X^{\prime\prime}$, then $\pi\circ\iota\circ\alpha^{\prime\prime}$ is a
morphism $A^{\prime\prime}\times P^{\prime\prime}\to B\times Q$ whose image
contains all points in $X$ that are not in the subset $Y$ parameterised by
$\alpha^{\prime}$. So assume that $(b,q)\in X(V)\subseteq B\times Q(V)$ does
not lie in $Y(V)$. Then there exists a linear map $\varphi:V\to U$ such that
$h(b,Q(\varphi)q)\neq 0$. Let $\psi\in\operatorname{Hom}\nolimits(V,U\oplus
V)$ be the map $v\mapsto(\varphi(v),v)$. Then, since
$(\operatorname{id}_{U}\oplus 0_{V,0})\circ\psi=\varphi$, the point
$q^{\prime}:=Q(\psi)q\in X(U\oplus V)$ satisfies
$h(b,Q(\operatorname{id}_{U}\oplus 0_{V,0})q^{\prime})=h(b,Q(\varphi)q)\neq
0$, i.e., $q^{\prime}$ lies in $(\mathrm{Sh}_{U}X)[1/h]$, which is the image
of $\iota\circ\alpha^{\prime\prime}$. Moreover, since
$(0_{U,0}\oplus\operatorname{id}_{V})\circ\psi=\operatorname{id}_{V}$, we have
$\pi_{V}(q^{\prime})=q$. Hence $q$ lies in the image of
$\pi\circ\iota\circ\alpha^{\prime\prime}$. This concludes the proof of
correctness of $\mathbf{parameterise}$. ∎

######  Lemma 3.5.1.

Let $Q$ be a pure polynomial functor and $X\subseteq B\times Q$ a closed
subset. Assume that $X$ is defined by its equations in $K[B\times Q(U)]$, let
$R$ be a subfunctor of $Q$, and set $Q^{\prime}:=Q/R$. Define $X^{\prime}$ as
the closure of the image of $X$ under the projection
$\operatorname{id}_{B}\times\pi:B\times Q\to B\times Q^{\prime}$. Assume that
$X(U)=(\operatorname{id}_{B}\times\pi_{U})^{-1}(X^{\prime}(U))$. Then
$X(V)=(\operatorname{id}_{B}\times\pi_{V})^{-1}(X^{\prime}(V))$ for all
$V\in\mathbf{Vec}$, so that $X=X^{\prime}\times R$, and moreover $X^{\prime}$
is defined by its equations in $K[B\times Q^{\prime}(U)]$.

###### Proof.

Let $(b,q)\in B\times Q(V)$ satisfy $(b,\pi_{V}(q))\in X^{\prime}(V)$. Then
for all $\varphi\in\operatorname{Hom}\nolimits(V,U)$ we have

$(b,\pi_{U}(Q(\varphi)(q)))=(b,Q^{\prime}(\varphi)(\pi_{V}(q)))\in
X^{\prime}(U)$

and hence, by assumption, $(b,Q(\varphi)(q))\in X(U)$. But then, since $X$ is
defined by its equations in $K[B\times Q(U)]$, we have $(b,q)\in X(V)$. This
proves the first statement.

For the last statement, suppose that $(b,q^{\prime})\in B\times Q^{\prime}(V)$
is such that $(b,Q^{\prime}(\varphi)(q^{\prime}))\in X^{\prime}(U)$ for all
$\varphi\in\operatorname{Hom}\nolimits(V,U)$. Pick any $q\in Q(V)$ with
$\pi_{V}(q)=q^{\prime}$. Then the same computation as above shows that
$(b,q)\in X(V)$, hence a fortiori $(b,q^{\prime})\in X^{\prime}(V)$. Hence
$X^{\prime}$ is defined by its equations in $K[B\times Q^{\prime}(U)]$. ∎

######  Lemma 3.5.2.

Let $Q$ be a pure polynomial functor, $R$ a (not necessarily irreducible)
subfunctor, $B$ an affine variety. Set $Y:=B\times Q$,
$Y^{\prime}:=B\times(Q/R)$. Let $X$ be a closed subset of $Y$ such that the
projection $\pi:Y\to Y^{\prime}$ restricts to an isomorphism from $X$ to a
closed subset $X^{\prime}$ of $Y^{\prime}$. Let $U$ be a vector space such
that $X$ is defined by its equations in $K[Y(U)]$. Then $X^{\prime}$ is
defined by its equations in $K[Y^{\prime}(U)]$.

###### Proof.

The idea is to find an inverse to $\pi$, i.e., a morphism $\psi:Y^{\prime}\to
Y$ such that

  1. (1)

$\pi_{V}\circ\psi_{V}=\operatorname{id}_{Y^{\prime}(V)}$

  2. (2)

$\psi|_{X^{\prime}}=(\pi|_{X})^{-1}$

Once we have found this $\psi$, we are done: Let $y^{\prime}\in
Y^{\prime}(V)$, such that for every linear map $\varphi:V\to U,$
$Y^{\prime}(\varphi)(y^{\prime})\in X^{\prime}(U)$, and hence, by property
(2), $\psi_{U}(Y^{\prime}(\varphi)(y^{\prime}))\in X(U)$. Since $\psi$ is a
morphism, we get

$\psi_{U}(Y^{\prime}(\varphi)(y^{\prime}))=(Y(\varphi)(\psi_{V}(y^{\prime}))\in
X(U).$

Since $X$ is defined by its equations in $K[Y(U)]$, we get that
$\psi_{V}(y^{\prime})\in X(V)$, and hence, with property $(1)$ of $\psi$,
$\pi_{V}(\psi_{V}(y^{\prime}))=y^{\prime}\in X^{\prime}(V)$.

By the Friedlander-Suslin-Lemma (Lemma 2.3.2 and proof of Lemma 2.7.3), it is
enough to find a $\operatorname{GL}\nolimits_{d}$-equivariant map
$\tilde{\psi}:Y^{\prime}(K^{d})\to Y(K^{d})$, where $d$ is the degree of $Q$,
that fulfills properties (1) and (2) (with $V=K^{d}$ and
$\psi_{K^{d}}=\tilde{\psi}$).

It is easy to find a possibly non-$\operatorname{GL}\nolimits_{d}$-equivariant
map $\hat{\psi}:Y^{\prime}(K^{d})\to Y(K^{d})$ with properties (1) and (2):
Note that $X^{\prime}(K^{d})$ is a closed subvariety of the affine space
${\mathbb{A}}^{n^{\prime}}_{B}$, where $n^{\prime}=\dim(Q(K^{d})/R(K^{d}))$
and $X(K^{d})$ is a closed subvariety of the affine space
${\mathbb{A}}^{n}_{B}$, where $n=\dim(Q(K^{d}))$. By basic properties of
affine spaces, the map $(\pi_{K^{d}}|_{X(K^{d})})^{-1}$ extends to a morphism
$\hat{\psi}$ from the ambient affine space ${\mathbb{A}}^{n^{\prime}}_{B}$ of
$X^{\prime}(K^{d})$ to the ambient affine space ${\mathbb{A}}^{n}_{B}$ of
$X(K^{d})$. Now $\hat{\psi}$ lives in some finite-dimensional space equipped
with a linear action of $\operatorname{GL}\nolimits_{d}$ given by
$(g,\eta)\mapsto Y(g)\circ\eta\circ Y^{\prime}(g^{-1})$. Since this space is
an algebraic group representation of $\operatorname{GL}\nolimits_{d}$ and
$\operatorname{GL}\nolimits_{d}$ is linearly reductive, there exists a
$\operatorname{GL}\nolimits_{d}$-equivariant linear map $R$ from the space
onto its subspace of $\operatorname{GL}\nolimits_{d}$-invariant elements—the
so-called Reynolds operator. Then $\tilde{\psi}:=R(\hat{\psi})$ is
$\operatorname{GL}\nolimits_{d}$-equivariant, and we claim that it still
satisfies (1) and (2).

By the Lefschetz principle, it is sufficient to check this when
$K={\mathbb{C}}$. In this case, $\tilde{\psi}$ takes the more explicit form

$\tilde{\psi}:=\int_{U_{d}({\mathbb{C}})}Y(\varphi)\circ\hat{\psi}\circ
Y^{\prime}(\varphi)^{-1}\,\mathrm{d}\varphi$

where
$U_{d}({\mathbb{C}})\subseteq\operatorname{GL}\nolimits_{d}({\mathbb{C}})$ is
the unitary group, and the integral is over the Haar measure (see e.g.
[Pro07]) with $\int_{U_{d}({\mathbb{C}})}\,\mathrm{d}\varphi=1$ (note that the
Haar integral needs to be taken over a compact group, so we have to use
$U_{d}({\mathbb{C}})$ instead of
$\operatorname{GL}\nolimits_{d}({\mathbb{C}})$).

Indeed, by construction, the morphism $\tilde{\psi}$ defined by this integral
is $U_{d}$-equivariant. To see that it is, indeed,
$\operatorname{GL}\nolimits_{d}$-equivariant, one uses that
$U_{d}({\mathbb{C}})$ is Zariski-dense in
$\operatorname{GL}\nolimits_{d}({\mathbb{C}})$ [Pro07, Section 8.6.1]. This is
a version of Weyl’s well-known unitary trick.

The following chain of equations now verifies that $\tilde{\psi}$ meets
property (1):

$\displaystyle\pi_{{\mathbb{C}}^{d}}\circ\tilde{\psi}=\pi_{{\mathbb{C}}^{d}}\circ\int_{U_{d}({\mathbb{C}})}Y(\varphi)\circ\hat{\psi}\circ
Y^{\prime}(\varphi)^{-1}\,\mathrm{d}\varphi=\int_{U_{d}({\mathbb{C}})}\pi_{{\mathbb{C}}^{d}}\circ
Y(\varphi)\circ\hat{\psi}\circ Y^{\prime}(\varphi)^{-1}\,\mathrm{d}\varphi$
$\displaystyle=\int_{U_{d}({\mathbb{C}})}Y^{\prime}(\varphi)\circ\pi_{{\mathbb{C}}^{d}}\circ\hat{\psi}\circ
Y^{\prime}(\varphi)^{-1}\,\mathrm{d}\varphi=\int_{U_{d}({\mathbb{C}})}\operatorname{id}_{Y^{\prime}({\mathbb{C}}^{d})}\,\mathrm{d}\varphi.$

For property (2), let $x\in X^{\prime}({\mathbb{C}}^{d})$. Then

$\displaystyle\tilde{\psi}(x)=\int_{U_{d}({\mathbb{C}})}Y(\varphi)(\hat{\psi}(\underbrace{Y^{\prime}(\varphi)^{-1}(x)}_{\in
X^{\prime}({\mathbb{C}}^{d})}))\,\mathrm{d}\varphi$
$\displaystyle=\int_{U_{d}({\mathbb{C}})}Y(\varphi)((\pi_{{\mathbb{C}}^{d}}|_{X({\mathbb{C}}^{d})})^{-1}(Y^{\prime}(\varphi)^{-1}(x)))\,\mathrm{d}\varphi$
$\displaystyle=\int_{U_{d}({\mathbb{C}})}Y(\varphi)(Y(\varphi)^{-1}((\pi_{{\mathbb{C}}^{d}}|_{X({\mathbb{C}}^{d})})^{-1}(x)))\,\mathrm{d}\varphi=(\pi_{{\mathbb{C}}^{d}}|_{X({\mathbb{C}}^{d})})^{-1}(x).$

Note that for the second to last equality we have used that $(\pi|_{X})^{-1}$
is a morphism, as is easy to verify. ∎

##  4\. Implicitisation

###  4.1. The result

The goal of this section is to prove the following more precise version of the
Main Theorem.

######  Theorem 4.1.1.

There exists an algorithm $\mathbf{implicitise}$ that, on input finite-
dimensional affine varieties $A,B$, pure polynomial functors $P,Q$, and a
morphism $\alpha:A\times P\to B\times Q$, computes a $U\in\mathbf{Vec}$ and
elements $f_{1},\ldots,f_{k}\in K[B\times Q(U)]$ that define the image closure
of $\alpha$, i.e., such that, for all $V\in\mathbf{Vec}$, we have

$\displaystyle\overline{\alpha(A(\overline{K})\times\overline{K}\otimes
P(V))}$ $\displaystyle=\\{(b,q)\in B(\overline{K})\times\overline{K}\otimes
Q(V)\mid\forall i\ \forall\varphi\in\operatorname{Hom}\nolimits_{K}(V,U):\
f_{i}(b,1\otimes Q(\varphi)q)=0\\}.$

###  4.2. The implicitisation algorithm

The algorithm $\mathbf{implicitise}(A,P,B,Q,\alpha)$ is run on countably many
parallel processors: the one where the original call is handled, plus
countably many labelled $0,1,2,3,\ldots$ where calls to an auxiliary procedure
$\mathbf{certify}$ is made. The latter has the following specification.

######  Proposition 4.2.1.

There exists a procedure that, on input finite-dimensional affine varieties
$A,A^{\prime},B$, pure polynomial functors $P,P^{\prime},Q$, and morphisms
$\alpha:A\times P\to B\times Q$ and $\alpha^{\prime}:A^{\prime}\times
P^{\prime}\to B\times Q$, has the following behaviour: if for all
$V\in\mathbf{Vec}$ we have

$\alpha^{\prime}(A^{\prime}(\overline{K})\times(\overline{K}\otimes
P^{\prime}(V)))\subseteq\overline{\alpha(A(\overline{K})\times(\overline{K}\otimes
P(V)))},$

then $\mathbf{certify}(B;Q;A;P;\alpha;A^{\prime};P^{\prime};\alpha^{\prime})$
terminates and returns “true”. Otherwise, it does not terminate.

The proof of Proposition 4.2.1 is deferred to §5. We can now present the steps
for $\mathbf{implicitise}(A,P,B,Q,\alpha)$.

  1. (1)

Set $n:=0$.

  2. (2)

While no instance of $\mathbf{certify}$ has returned “true”, perform steps
(3)–(6):

  3. (3)

Set $U:=K^{n}$ and, by classical elimination, compute defining equations
$f_{n,1},\ldots,f_{n,k_{n}}\in K[B\times Q(U)]$ for the image closure of
$\alpha_{U}$
(this is a finite-dimensional affine variety).

  4. (4)

Compute
$(A_{n},P_{n},\alpha_{n}):=\mathbf{parameterise}(B;Q;U;f_{n,1},\ldots,f_{n,k_{n}})$.

  5. (5)

On the $n$-th processor, start
$\mathbf{certify}(B;Q;A;P;\alpha;A_{n};P_{n};\alpha_{n})$.

  6. (6)

Set $n:=n+1$.

  7. (7)

if the $m$-th processor has returned “true”, then return
$(K^{m};f_{m,1},\ldots,f_{m,k_{m}})$.

###  4.3. Correctness and termination of $\mathbf{implicitise}$

###### Proof of Theorem 4.1.1.

By Corollary 2.4.3, there exists a value of $n$ such that the tuple
$(K^{n};f_{n,1},\ldots,f_{n,k_{n}})$ computed in iteration $n$ is a correct
output for $\mathbf{implicitise}$. It then follows that $\alpha_{n}$, which,
by virtue of $\mathbf{parameterise}$, parameterises the closed subset of
$B\times Q$ defined by $f_{n,1},\ldots,f_{n,k_{n}}$, has its image contained
in the closure of the image of $\alpha$. Hence, by Proposition 4.2.1, the
$n$-th call to $\mathbf{certify}$ terminates and returns “true”. This shows
that $\mathbf{implicitise}$ terminates.

Next, when it terminates with output $(K^{m};f_{m,1},\ldots,f_{m,k_{m}})$,
then this is because the image of $\alpha_{m}$, which equals the closed subset
of $B\times Q$ defined by $f_{m,1},\ldots,f_{m,k_{m}}$, is contained in the
image closure of $\alpha$. Since, conversely, the image closure of $\alpha$ is
contained in the closed subset defined by $f_{m,1},\ldots,f_{m,k_{m}}$, the
output is correct. ∎

##  5\. Certifying inclusion of image closures

###  5.1. An instructive example

######  Example 5.1.1.

Let $\alpha:(S^{1})^{2}\to S^{3}$ be the morphism defined by
$\alpha(u,v)=u^{3}+v^{3}$. The image closure of $\alpha$ is the set of
symmetric three-tensors of border Waring rank at most $2$. On the other hand,
let $\beta:(S^{1})^{2}\to S^{3}$ be the morphism defined by
$\beta(u,v)=6u^{2}v$. Then we have

$\beta(u,v)=6u^{2}v=\lim_{t\to
0}[(t^{2}v+t^{-1}u)^{3}+(t^{2}v-t^{-1}u)^{3}]=\lim_{t\to
0}\alpha(t^{2}v+t^{-1}u,t^{2}v-t^{-1}u)$

and this implies that
$\operatorname{im}(\beta)\subseteq\overline{\operatorname{im}(\alpha)}$.
However,
$\operatorname{im}(\beta)\not\subseteq\operatorname{im}(\alpha)$—indeed, it is
well known that the Waring rank of cubics of the form $u^{2}v$ with $u,v$
linearly independent vectors is equal to $3$.

This example shows a certificate for
$\operatorname{im}\beta\subseteq\overline{\operatorname{im}\alpha}$, namely,
that $\beta$ is a limit of a composition $\alpha\circ\gamma_{t}$ with

$\gamma_{t}:(S^{1})^{2}\to(S^{1})^{2},(u,v)\mapsto(t^{2}v+t^{-1}u,t^{2}v-t^{-1}u).$

Roughly speaking, whenever $\operatorname{im}(\beta)$ is contained in
$\overline{\operatorname{im}(\alpha)}$, where $\beta,\alpha$ are morphisms
into $B\times Q$, with $B$ a finite-dimensional variety and $Q$ a pure
polynomial functor, there is a certificate of this inclusion such as the one
above—see below for the precise statement. However, we are not aware of any a
priori lower bound on the (negative) exponents of $t$ in such a certificate.
This is why, in Proposition 4.2.1, the procedure $\mathbf{certify}$ does not
terminate when no certificate exists.

###  5.2. An excursion to infinite dimensions

We collect some material on $\operatorname{GL}\nolimits$-varieties. The
results stated here appear in [BDES21, BDES22] or can directly be derived from
results there.

Given a finite-dimensional variety $A$ and a pure polynomial functor $P$, we
construct the inverse limit $\lim_{\leftarrow n}(A\times P(K^{n}))=A\times
P_{\infty}$, where the projections $P(K^{n+1})\to P(K^{n})$ are of the form
$P(\pi)$ with $\pi$ the standard projection $K^{n+1}\to K^{n}$. Rather than as
a set of $K$-valued points, we will regard $A\times P_{\infty}$ as a reduced,
affine $K$-scheme, namely, the spectrum of the ring $K[A]\otimes_{K}R$, where
$R$ is the symmetric algebra of the countable-dimensional vector space
$\lim_{n\to\infty}P(K^{n})^{*}$. The group
$\operatorname{GL}\nolimits:=\bigcup_{n=0}^{\infty}\operatorname{GL}\nolimits_{n}(K)$
acts on $A\times P_{\infty}$ by means of automorphisms and $A\times
P_{\infty}$ is a $\operatorname{GL}\nolimits$-variety in the sense of
[BDES21]. More generally, if $X$ is a closed subset of a polynomial functor,
then the inverse limit $X_{\infty}$ of all $X(K^{n})$ is a
$\operatorname{GL}\nolimits$-variety. The association $X\mapsto X_{\infty}$ is
an equivalence of categories with the category of affine
$\operatorname{GL}\nolimits$-varieties, which sends a morphism $\alpha:X\to Y$
to a $\operatorname{GL}\nolimits$-equivariant morphism
$\alpha_{\infty}:X_{\infty}\to Y_{\infty}$ of affine schemes over $K$.

Let $\alpha:A\times P\to B\times Q$ and $\alpha^{\prime}:A^{\prime}\times
P^{\prime}\to B\times Q$ be morphisms as in Proposition 4.2.1. Then the
following two statements are equivalent:

  1. (1)

$\overline{\operatorname{im}(\alpha_{\infty})}\supseteq\operatorname{im}(\alpha^{\prime}_{\infty})$
and

  2. (2)

$\overline{\operatorname{im}(\alpha_{V})}\supseteq\operatorname{im}(\alpha^{\prime}_{V})$
for all $V\in\mathbf{Vec}$.

It is (2) which we want to certify in
$\mathbf{certify}(B;Q;A;P;\alpha;A^{\prime},P^{\prime},\alpha^{\prime})$. From
now on, we assume that $A,A^{\prime}$ are irreducible—in the procedure
$\mathbf{certify}$ we will reduce to this case.

Now [BDES22] contains the following useful criterion for (1). Let $a^{\prime}$
be the generic point of $A^{\prime}$ and let $p^{\prime}\in
P^{\prime}_{\infty}(K)$ be a point whose $\operatorname{GL}\nolimits$-orbit is
dense in $P_{\infty}$ (such points exist, see [BDES21]). Let
$x:=\alpha^{\prime}_{\infty}(a^{\prime},p^{\prime})$; this is an
$\Omega$-point of $B\times Q_{\infty}=:X_{\infty}$, where
$\Omega=K(A^{\prime})$, and the $\operatorname{GL}\nolimits$-orbit of $x$ is
dense in $\operatorname{im}(\alpha^{\prime}_{\infty})$. Write
$Y_{\infty}:=A\times P_{\infty}$.

######  Theorem 5.2.1.

We have
$\operatorname{im}(\alpha^{\prime}_{\infty})\subseteq\overline{\operatorname{im}(\alpha_{\infty})}$
if and only if there exists a finite-dimensional field extension
$\widetilde{\Omega}$ of $\Omega$ and a bounded $\widetilde{\Omega}((t))$-point
$y(t)$ of $Y_{\infty}$ such that $\lim_{t\to 0}\alpha_{\infty}(y(t))=x$.

This follows from [BDES22, Theorem 5.8] and its proof. Here “bounded” means
that the exponents of $t$ in the countably many coordinates of the
$P_{\infty}$-component of $y(t)$ are uniformly bounded from below. This
ensures that $\alpha_{\infty}(y(t))$ is a well-defined
$\widetilde{\Omega}((t))$-point of $X_{\infty}$. The theorem says that we can
choose $y(t)$ such that $\alpha_{\infty}(y(t))$ is in fact an
$\widetilde{\Omega}[[t]]$-point of $X_{\infty}$ and that setting $t$ to zero
yields $x$.

The procedure $\mathbf{certify}$ should certify the existence of $y(t)$. To
this end, we will narrow down the space in which to search for $y(t)$ to an
increasing chain of finite-dimensional varieties. First, we will show that
$y(t)$ needs not use more of $P_{\infty}$ than can be obtained by applying
morphisms to $p^{\prime}$. To do so (see Proposition 5.4.1 below), we now
introduce systems of variables in Schur functors.

###  5.3. Systems of variables in Schur functors

Fix a field extension $\Omega$ of $K$. For every nonempty partition $\lambda$,
$S_{\lambda,\infty}$ is an affine scheme over $K$ whose $\Omega$-valued points
form an $\Omega$-vector space of uncountable dimension. Let
$V_{\lambda}\subseteq S_{\lambda,\infty}(\Omega)$ be the set of all points $s$
for which there exist $k$, partitions $\mu_{1},\ldots,\mu_{k}$ with
$0<|\mu_{i}|<|\lambda|$, an $\Omega$-valued point $\alpha$ of
$\mathbf{Map}(S_{\mu_{1}}\oplus\cdots\oplus S_{\mu_{k}},S_{\lambda})$, and an
$\Omega$-valued point $p$ of $S_{\mu_{1},\infty}\times\cdots\times
S_{\mu_{k},\infty}$ such that $\alpha_{\infty}(p)=s$. (Recall the definition
of $\mathbf{Map}(P^{\prime},P)$ from Remark 2.7.4.)

######  Example 5.3.1.

If $\lambda=(2)$, then $S_{\lambda,\infty}(\Omega)$ is the space of infinite-
by-infinite symmetric matrices with entries in $\Omega$, and $V_{\lambda}$ is
the subspace of matrices of finite rank.

Now $V_{\lambda}$ is a proper $\Omega$-vector subspace of
$S_{\lambda,\infty}(\Omega)$, and we choose any $\Omega$-basis
$(\xi_{\lambda,i})_{i\in I_{\lambda}}$ of a vector space complement to
$V_{\lambda}$ in $S_{\lambda,\infty}(\Omega)$, where $I_{\lambda}$ is an
uncountable index set. We call the $\xi_{\lambda,i}$ variables. We choose
these variables for every $\lambda$ and write $\xi$ for the resulting
uncountable tuple; $\xi$ is what we call a system of variables (over $\Omega$)
for all Schur functors. If $f$ is an $\Omega$-valued point of
$\mathbf{Map}(S_{\mu_{1}}\oplus\cdots\oplus S_{\mu_{k}},S_{\lambda})$ and we
fix indices $i_{1}\in I_{\mu_{1}},\ldots,i_{k}\in I_{\mu_{k}}$, then we will
write $f(\xi)$ for $f(\xi_{\mu_{1},i_{1}},\ldots,\xi_{\mu_{k},i_{k}})$. This
is slight abuse of notation, since it is not apparent from the formula
$f(\xi)$ which indices were chosen, but the notation is compatible with the
notation $f(x)$ for a polynomial that uses finitely many of an uncountable set
of variables $x$.

######  Remark 5.3.2.

We need to fix the extension $\Omega$ first and then choose the system of
variables. It is not true that a system of variables chosen over $K$ is also a
system of variables over field extensions of $K$, as
$S_{\lambda,\infty}(K)\otimes_{K}\Omega=S_{\lambda,\infty}(\Omega)$ only when
$\Omega$ is finite-dimensional over $K$.

The following proposition follows readily from the material in [BDES21].

######  Proposition 5.3.3.

Let $S=S_{\mu_{1}}\oplus\cdots\oplus S_{\mu_{k}}$ be a pure polynomial
functor, and let $s=(s_{1},\ldots,s_{k})\in S_{\infty}(\Omega)$. Then the
$s_{i}$ can be chosen as part of a system of variables if and only if the
$\operatorname{GL}\nolimits$-orbit of $s$ is dense in $S_{\infty}(\Omega)$.

The following theorem expresses that the variables in a system are independent
and generate all vectors in all Schur functors.

######  Theorem 5.3.4.

Fix a field extension $\Omega$ of $K$ and a system of variables $\xi$ over
$\Omega$ for all Schur functors. Then for every nonempty partition $\lambda$
and any $p\in S_{\lambda,\infty}(\Omega)$, there exist partitions
$\lambda_{1},\ldots,\lambda_{k}$ and an $\Omega$-valued point $f$ of
$\mathbf{Map}(S_{\lambda_{1}}\oplus\cdots\oplus S_{\lambda_{k}},S_{\lambda})$
and variables $\xi_{\lambda_{j},i_{j}}$ for $j=1,\ldots,k$ and $i_{j}\in
I_{\lambda_{j}}$ such that
$p=f_{\infty}(\xi_{\lambda_{1},i_{1}},\ldots,\xi_{\lambda_{k},i_{k}})$.
Moreover, if $f$ really depends on all $\xi_{\lambda_{j},i_{j}}$ in the sense
that replacing one of them by zero changes the outcome, then, up to
permutations of $\\{1,\ldots,k\\}$, the partitions $\lambda_{j}$, the
variables $\xi_{\lambda_{j},i_{j}}$, and $f$ are unique.

###### Proof.

The existence of $f$ follows by induction on $|\mu|$: we may write $p$

$p=c_{1}\xi_{\lambda,i_{1}}+c_{2}\xi_{\lambda,i_{2}}+\cdots+c_{l}\xi_{\lambda,i_{l}}+\tilde{p}$

with $\tilde{p}\in V_{\lambda}$ and (unique) $i_{1},\ldots,i_{l}\in
I_{\lambda}$ and $c_{1},\ldots,c_{l}\in\Omega$. Now
$\tilde{p}=\alpha_{\infty}(q)$, for a suitable $\alpha$, where $q$ is an
$\Omega$-valued point of $S_{\mu_{1},\infty}\times\cdots\times
S_{\mu_{k},\infty}$ with $0<|\mu_{i}|<|\lambda|$ for all $i$. By the induction
hypothesis, the components $q_{i},i=1,\ldots,k$ of $q$ are of the form
$f_{i,\infty}(\xi)$, and then $p$ equals

$\alpha_{\infty}(f_{1,\infty}(\xi),\ldots,f_{k,\infty}(\xi))+\sum_{i=1}^{l}c_{i}\xi_{\lambda,i},$

so that
$f:=\alpha(f_{1},\ldots,f_{k})+\sum_{i=1}^{l}c_{i}\operatorname{id}_{S_{\lambda}}$
does the trick for the obvious choice of variables.

For uniqueness, it suffices to show that if $f$ is a nonzero $\Omega$-valued
point of $\mathbf{Map}(S_{\mu_{1}}\oplus\ldots\oplus S_{\mu_{k}},S_{\lambda})$
and $\xi_{\mu_{1},i_{1}},\ldots,\xi_{\mu_{k},i_{k}}$ are distinct variables,
then $f_{\infty}(\xi)\neq 0$. Indeed, by Proposition 5.3.3, the
$\operatorname{GL}\nolimits$-orbit of
$(\xi_{\mu_{1},1},\ldots,\xi_{\mu_{k},i_{k}})$ is dense, and $f_{\infty}$ is
$\operatorname{GL}\nolimits$-equivariant, so that $f_{\infty}(\xi)=0$ implies
that $f=0$, a contradiction. ∎

######  Remark 5.3.5.

Our notion of systems of variables is inspired by [ESS19]. Indeed, If we
restrict a system of variables in Schur functors to the symmetric powers
$S^{d}$ with $d=0,1,\ldots$, then the proof of [ESS19, Theorem 1.1] shows that
these are algebraically independent (over $\Omega$) generators of the ring
$\bigoplus_{d}S^{d}_{\infty}(\Omega)$, and that this ring is therefore
abstractly isomorphic to a polynomial ring. This fact can also be deduced from
Theorem 5.3.4 above.

###  5.4. Narrowing down the search for $y(t)$

In this section, we retain the notation from §5.2. The following diagram
represents the situation:

$\textstyle{(a^{\prime},p^{\prime})\in A^{\prime}\times
P^{\prime}_{\infty}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{\prime}_{\infty}}$$\textstyle{y(t)=(a(t),p(t))\in(A\times
P_{\infty})(\widetilde{\Omega}((t)))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\quad\alpha_{\infty}}$$\textstyle{x=(b,q)\in(B\times
Q_{\infty})(\Omega)}$

where $\Omega=K(A^{\prime})$, $a^{\prime}$ is the generic point of
$A^{\prime}$, $p^{\prime}$ is a $K$-point of $P_{\infty}$ with dense
$\operatorname{GL}\nolimits$-orbit, and we want to certify the existence of
$y(t)$, defined over a finite extension $\widetilde{\Omega}$ of $\Omega$, such
that $\lim_{t\to 0}\alpha_{\infty}(y(t))=x$.

######  Proposition 5.4.1.

If a $y(t)$ as in Theorem 5.2.1 exists, then it can be chosen of the form
$(a(t),\gamma(t)_{\infty}(p^{\prime}))$ with $a(t)\in
A(\widetilde{\Omega}((t)))$ and $\gamma(t)$ a
$\widetilde{\Omega}[t,t^{-1}]$-valued point of the finite-dimensional affine
space $\mathbf{Map}(P^{\prime},P)$.

###### Proof.

Write $y(t)=(a(t),p(t))$. First, terms in $p(t)$ of sufficiently high degree
in $t$ do not contribute to $\lim_{t\to 0}\alpha_{\infty}(a(t),p(t))$, so we
may truncate $p(t)$ and assume that it is a finite sum
$\sum_{d=m_{1}}^{m_{2}}t^{d}p_{d}$ where each $p_{d}\in
P_{\infty}(\widetilde{\Omega})$.

Now write $P^{\prime}=S_{\lambda_{1}}\oplus\cdots\oplus S_{\lambda_{k}}$,
where the $\lambda_{i}$ are partitions. Accordingly, decompose
$p^{\prime}=(p^{\prime}_{1},\ldots,p^{\prime}_{k})$ with $p^{\prime}_{i}\in
S_{\lambda_{i},\infty}$. Over the field extension $\widetilde{\Omega}$ of $K$,
choose a system of variables $\xi$ in such a manner that
$p^{\prime}_{1},\ldots,p^{\prime}_{k}$ are among these variables; this can be
done by Proposition 5.3.3 because $\operatorname{GL}\nolimits\cdot p^{\prime}$
is dense in $P^{\prime}_{\infty}$. Then, by Theorem 5.3.4, we have
$p_{d}=f_{d,\infty}(\xi)$ for all $d$, where $f_{d}$ is an (essentially
unique) morphism into $P$ with coefficients in $\widetilde{\Omega}$.

Recall from Remark 2.7.5 that $\alpha$ splits as $\alpha^{(0)}:A\to B$ and
$\alpha^{(1)}:A\times P\to Q$, and similarly for $\alpha^{\prime}$. The limit
$\lim_{t\to 0}\alpha^{(1)}_{\infty}(a(t),p(t))$ equals

$g_{\infty}(p_{m_{1}},\ldots,p_{m_{2}})=g_{\infty}(f_{m_{1},\infty}(\xi),\ldots,f_{m_{2},\infty}(\xi))$

for some $\widetilde{\Omega}$-point $g$ of
$\mathbf{Map}(P^{m_{2}-m_{1}+1},Q)$. On the other hand, by the choice of
$y(t)$, it equals $(\alpha^{\prime})^{(1)}_{\infty}(p^{\prime})$. In the
latter expression, only the variables $p^{\prime}_{1},\ldots,p^{\prime}_{k}$
appear. By the uniquess statement in Theorem 5.3.4, the same must apply to
$g_{\infty}(f_{m_{1},\infty}(\xi),\ldots,f_{m_{2},\infty}(\xi))$.

Therefore, replacing each $p_{d}$ by
$\tilde{p}_{d}:=f_{d,\infty}(p^{\prime},0)$, where all variables not among the
variables $p_{1}^{\prime},\ldots,p_{k}^{\prime}$ are set to zero, yields a
$\tilde{y}(t)$ with the same property as $y(t)$: $\lim_{t\to
0}\alpha_{\infty}(\tilde{y}(t))=x$. Now
$\gamma(t):=\sum_{d}t^{d}f_{d}(\cdot,0)$ is the desired
$\widetilde{\Omega}[t,t^{-1}]$-valued point of $\mathbf{Map}(P^{\prime},P)$. ∎

Recall from Remark 2.7.5 that $(\alpha^{\prime})^{(1)}$ can be regarded an
$\Omega$-point of $\mathbf{Map}(P^{\prime},Q)$. Similarly,
$\alpha^{(1)}(a(t),\cdot)$ can be regarded an $\widetilde{\Omega}((t))$-point
of $\mathbf{Map}(P,Q)$.

######  Lemma 5.4.2.

A point $(a(t),\gamma(t)_{\infty}(p^{\prime}))$ as in Proposition 5.4.1
satisfies the property $\lim_{t\to
0}\alpha_{\infty}(a(t),\gamma(t)_{\infty}(p^{\prime}))=\alpha^{\prime}(a^{\prime},p^{\prime})=:(b,q)\in(B\times
Q_{\infty})(\Omega)$ if and only if, first, $\lim_{t\to
0}\alpha^{(0)}(a(t))=b$ and, second, the $\widetilde{\Omega}((t))$-point
$\alpha^{(1)}(a(t),\cdot)\circ\gamma(t)$ of $\mathbf{Map}(P^{\prime},Q)$
satisfies

$\lim_{t\to 0}\alpha^{(1)}(a(t),\cdot)\circ\gamma(t)=(\alpha^{\prime})^{(1)};$

an equality of $\widetilde{\Omega}$-points in $\mathbf{Map}(P^{\prime},Q)$.

###### Proof.

The statement “if” is immediate, by substituting $p^{\prime}$; and the
statement “only if” follows from the fact that the
$\operatorname{GL}\nolimits$-orbit of $p^{\prime}$ is dense in
$P^{\prime}_{\infty}$. ∎

###  5.5. Greenberg’s approximation theorem

We have almost arrived at a countable chain of finite-dimensional varieties in
which we can look for $y(t)$. The only problem is that the point $a(t)\in
A(\widetilde{\Omega}((t)))$ does not yet have a finite representation. For
concreteness, assume that $A$ is given by a prime ideal
$I=(f_{1},\ldots,f_{r})$ in $K[x_{1},\ldots,x_{m}]$, $B$ is embedded in
$K^{n}$, and $\alpha^{(0)}:A\to B$ is the restriction of some polynomial map
$\alpha^{(0)}:K^{m}\to K^{n}$.

Then $a(t)$ is an $m$-tuple in $\widetilde{\Omega}((t))^{m}$, and together
with $\gamma(t)$ it is required to satisfy the following properties from Lemma
5.4.2:

  1. (i)

$f_{i}(a(t))=0$ for $i=1,\ldots,r$;

  2. (ii)

$\lim_{t\to 0}\alpha^{(0)}(a(t))=b$;

  3. (iii)

and $\lim_{t\to
0}\alpha^{(1)}(a(t),\cdot)\circ\gamma(t)=(\alpha^{\prime})^{(1)}$.

Suppose that we fix a lower bound $-d_{1}$, with $d_{1}\in{\mathbb{Z}}_{\geq
0}$, on the exponents of $t$ appearing in $a(t)$ or in $\gamma(t)$. From the
data of $\alpha$ and $d_{1}$, one can compute a bound
$d_{2}\in{\mathbb{Z}}_{\geq 0}$ such that the validity of (ii) and (iii) do
not depend on the terms in $a(t)$ or $\gamma(t)$ with exponents $>d_{2}$.
However, (i) does depend on all (typically infinitely many) terms of $a(t)$.
Here Greenberg’s approximation theorem comes to the rescue. As this theorem
requires formal power series rather than Laurent series, we put
$\tilde{a}(t):=t^{d_{1}}a(t)$. Accordingly, replace each $f_{i}$ by
$\tilde{f}_{i}:=t^{e}f_{i}(t^{-d_{1}}x_{1},\ldots,t^{-d_{1}}x_{n})$ where $e$
is large enough such that all coefficients of $\tilde{f}_{i}$ for all $i$ are
in $\widetilde{\Omega}[[t]]$. Note that $a(t)$ is a root of all $f_{i}$ if and
only if $\tilde{a}(t)$ is a root of all $\tilde{f}_{i}$.

######  Theorem 5.5.1 (Greenberg, [Gre66]).

There exists numbers $N_{0}\geq 1,c\geq 1,s\geq 0$ such that for all $N\geq
N_{0}$ and $\overline{a}(t)\in\widetilde{\Omega}[[t]]^{n}$ with
$\tilde{f}_{i}(\overline{a}(t))\equiv 0\mod t^{N}$ for all $i=1,\ldots,r$
there exists an $\tilde{a}(t)\in\widetilde{\Omega}[[t]]^{n}$ such that
$\tilde{a}(t)\equiv\overline{a}(t)\mod t^{\lceil\frac{N}{c}\rceil-s}$ and
moreover $f_{i}(\tilde{a}(t))=0$ for all $i$. Moreover, $N_{0},c,s$ can be
computed from $\tilde{f}_{1},\ldots,\tilde{f}_{r}$.

As a matter of fact, the computability, which is crucial to our work, is only
implicit in [Gre66]; it is made explicit in the overview paper [Ron18].

######  Corollary 5.5.2.

There exist natural numbers $d_{2},N_{1}$, which can be computed from $d_{1}$
and $f_{1},\ldots,f_{r}$, $\alpha$, such that the following statements are
equivalent:

  1. (1)

a pair $a(t),\gamma(t)$ with properties (i)–(iii) exists that has no exponents
of $t$ smaller than $-d_{1}$;

  2. (2)

a pair $a(t),\gamma(t)$ exists with all exponents of $t$ in the interval
$\\{-d_{1},\ldots,d_{2}\\}$ that satisfies (ii) and (iii), and that satisfies
(i) modulo $t^{N_{1}}$.

###### Proof.

The implication (1) $\Rightarrow$ (2) holds for any choice of $N_{1}$ if
$d_{2}$ is chosen large enough so that the terms of $a(t),\gamma(t)$ with
degree $>d_{2}$ in $t$ do not affect (ii),(iii), and do not contribute to the
terms of degree $<N_{1}$ in $f_{i}(a(t))$ for any $i$.

For the converse, first we compute $\tilde{f}_{i}$ and $e$ as above; they
depend on the choice of $d_{1}$. Then we compute $N_{0},c,s$ as in Greenberg’s
theorem. Compute $N_{1}\geq N_{0}-e$ such that terms in $a(t),\gamma(t)$ in
which $t$ has exponent at least $\lceil\frac{N_{1}+e}{c}\rceil-s-d_{1}$ do not
affect properties (ii)–(iii), and then compute $d_{2}$ as in the first
paragraph.

Given a pair $a(t),\gamma(t)$ as in (2), set $\bar{a}(t):=t^{d_{1}}a(t)$.
Then, for each $i$,

$\tilde{f}_{i}(\bar{a}(t))=t^{e}f_{i}(a(t))\equiv 0\mod t^{N_{1}+e}.$

Then, since $N_{1}+e\geq N_{0}$, Greenberg’s theorem yields
$\tilde{a}(t)\in\widetilde{\Omega}[[t]]^{n}$ such that
$\tilde{f}_{i}(\tilde{a}(t))=0$ for all $i$ and such that

$\tilde{a}(t)=\bar{a}(t)\mod t^{\lceil\frac{N_{1}+e}{c}\rceil-s}.$

Now set $a_{1}(t):=t^{-d_{1}}\tilde{a}(t)$, so that $f_{i}(a_{1}(t))=0$ for
all $i$—this is property (i)—and

$a_{1}(t)\equiv a(t)\mod t^{\lceil\frac{N_{1}+e}{c}\rceil-s-d_{1}}.$

Since the terms of $a(t)$ with exponent of degree at least
$\lceil\frac{N_{1}+e}{c}\rceil-s-d_{1}$ do not affect (ii) and (iii), the pair
$a_{1}(t),\gamma(t)$ also satisfy these conditions. ∎

###  5.6. The procedure $\mathbf{certify}$

To compute
$\mathbf{certify}(B;Q;A;P;\alpha;A^{\prime};P^{\prime};\alpha^{\prime})$, we
proceed as follows. For convenience, we again assume that we have sufficiently
many processors working in parallel.

  1. (1)

If $A$ and $A^{\prime}$ are not both irreducible, decompose $A$ into
irreducible components $A_{i}$ and $A^{\prime}$ into irreducible components
$A^{\prime}_{j}$, and assign the computation of
$\mathbf{certify}(B;Q;A_{i};P;\alpha|_{A_{i}\times
P};A_{j}^{\prime},P^{\prime},\alpha^{\prime}_{A^{\prime}_{j}\times
P^{\prime}})$ for all $i,j$ to distinct processors. As soon as for each $j$
there exists at least one $i$ such that the computation returns “true”, return
“true”.

(So in what follows we may assume that $A,A^{\prime}$ are irreducible. They
are given by prime ideals $I\subseteq K[x_{1},\ldots,x_{n}]$ and $J\subseteq
K[y_{1},\ldots,y_{m}]$, respectively.)

  2. (2)

Let $f_{1},\ldots,f_{r}$ be generators of $I$.

  3. (3)

Compute $b:=(\alpha^{\prime})^{(0)}(a^{\prime})$ where $a^{\prime}$ is the
generic point of $A^{\prime}$.

So $a^{\prime}$ is just the $m$-tuple $(y_{1}+J,\ldots,y_{m}+J)\in\Omega^{m}$,
where $\Omega$ is the fraction field of $K[y_{1},\ldots,y_{m}]/J$.

  4. (4)

Compute the $\Omega$-valued point $(\alpha^{\prime})^{(1)}$ of
$\mathbf{Map}(P^{\prime},Q)$.

  5. (5)

Construct a $K$-basis $\gamma_{1},\ldots,\gamma_{m}$ of the vector space
$\mathbf{Map}(P^{\prime},P)$.

  6. (6)

Set $d_{1}:=0$, $r:=$“false”.

  7. (7)

While not $r$, perform the steps (8)–(10):

  8. (8)

From $\alpha$ and $d_{1}$, compute the natural numbers $N_{1},d_{2}$ from
Corollary 5.5.2 and make the Ansatz
$\gamma(t)=\sum_{i=1}^{m}c_{i}(t)\gamma_{m}$ where $c_{i}(t)$ is a linear
combination of $t^{-d_{1}},\ldots,t^{d_{2}}$ with coefficients to be
determined in an extension of $\Omega$; and the Ansatz
$a(t)=(a_{1}(t),\ldots,a_{n}(t))$, where $a_{i}$ is also a linear combination
of $t^{-d_{1}},\ldots,t^{d_{2}}$ with coefficients to be determined.

  9. (9)

The desired properties of $(a(t),\gamma(t))$ from the second item of Corollary
5.5.2 translate into a system of polynomial equations for the
$(m+n)\cdot(d_{2}+d_{1}+1)$ coefficients of the $c_{i}(t)$ and the $a_{i}(t)$.
By a Gröbner basis computation, test whether a solution exists over an
algebraic closure of $\Omega$. If so, set $r:=$“true”.

  10. (10)

Set $d_{1}:=d_{1}+1$.

  11. (11)

Return “true”.

###### Proof of Proposition 4.2.1.

The first step is justified by the observation that the image closure of
$\alpha$ contains the image of $\alpha^{\prime}$ if and only if for each $j$,
the image of $\alpha^{\prime}|_{A^{\prime}_{j}\times P^{\prime}}$ is contained
in the image closure of some $\alpha|_{A_{i}\times P}$.

If the image closure of $\alpha$ contains the image of $\alpha^{\prime}$, then
by Theorem 5.2.1, Proposition 5.4.1, Lemma 5.4.2, and Corollary 5.5.2, the
procedure $\mathbf{certify}$ terminates and returns “true”. Otherwise, by the
same results, the system of equations in step (9) does not have a solution,
and the procedure does not terminate. ∎

##  6\. Conclusion

The Noetherianity theorem [Dra19] and the unirationality theorem [BDES21]
imply that closed subsets of polynomial functors admit two different
descriptions: an implicit description by finitely many equations, and a
description as the image (closure) of some morphism. In this paper, we have
described highly nontrivial algorithms $\mathbf{parameterise}$ and
$\mathbf{implicitise}$ that translate between these two descriptions.
Interestingly, $\mathbf{implicitise}$ requires calls to
$\mathbf{parameterise}$ as well as Greenberg’s approximation theorem for a
stopping criterion. We do not expect these two algorithms to be implemented in
full generality anytime soon, but we do believe that their structure may guide
us in finding equations for closed subsets of polynomial functors such as
“cubics of q-rank $\leq 2$”.

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