# Generic differential equations are strongly minimal

Matthew DeVilbiss  and  James Freitag  Matthew DeVilbiss, University of
Illinois Chicago, Department of Mathematics, Statistics, and Computer Science,
851 S. Morgan Street, Chicago, IL, USA, 60607-7045<EMAIL_ADDRESS>James
Freitag, University of Illinois Chicago, Department of Mathematics,
Statistics, and Computer Science, 851 S. Morgan Street, Chicago, IL, USA,
60607-7045<EMAIL_ADDRESS>
###### Abstract.

In this manuscript we develop a new technique for showing that a nonlinear
algebraic differential equation is strongly minimal based on the recently
developed notion of the degree of nonminimality of Freitag and Moosa. Our
techniques are sufficient to show that generic order $h$ differential
equations with nonconstant coefficients are strongly minimal, answering a
question of Poizat (1980).

###### 2020 Mathematics Subject Classification:

03C45, 14L30, 12H05, 32J99

The authors were partially supported by NSF CAREER award 1945251 during the
course of this work. The authors thank Dave Marker, Ronnie Nagloo, Anand
Pillay, and especially Rahim Moosa for useful conversations around this work.
The techniques developed in this paper build on the thesis of Jonathan Wolf
cited below.

##  1\. Introduction

Let $f(x)=0$ be an algebraic differential equation of in a single
indeterminate $x$ with coefficients in a differential field $(K,\delta)$ of
characteristic zero. In this manuscript, we are particularly interested in the
case that $f(x)$ is nonlinear and of order $\geq 2$. The central property we
study is the _strong minimality_ of the solution set of $f(x)=0$. The notion
of strong minimality comes from model theory; in general, a _definable set_
$X$ is _strongly minimal_ if every definable subset is finite or cofinite,
uniformly in parameters. In our setting, we are interested in the situation
$X=\\{x\in\mathcal{U}\,|\,f(x)=0\\}$ is the set of solutions to an algebraic
differential equation where $\mathcal{U}$ is a differentially closed field.
Let $h$ be the order of $f$ – that is, the highest derivative of $x$ appearing
in $f$. The strong minimality of $X$ is equivalent to:

  * •

$f$ is irreducible as a (multivariate) polynomial over $K^{alg}$ and given any
$a\in\mathcal{U}$ with $f(a)=0,$ and any differential field
$K_{1}\leq\mathcal{U}$ with $K\leq K_{1}$, the transcendence degree of
$K_{1}\langle a\rangle$ over $K_{1}$ is either $0$ or $h$.

Strong minimality is an intensively studied property of definable sets, and
has been at the center of many important number theoretic applications of
model theory and differential algebra [4, 11, 12, 20]. Despite this, there are
relatively few (classes of) equations which have been shown to satisfy the
property - so few, that we are in fact able to give below what we believe to
be a (at the moment) comprehensive list of those equations which have been
shown to be strongly minimal. Showing the strong minimality of a given
equation is itself sometimes a motivational goal, but often it is an important
piece of a more elaborate application, since it allows one to use powerful
tools from geometric stability theory. The existing strategies to prove strong
minimality are widely disparate but apply only to very special cases. In
roughly chronological order:

  1. (1)

Poizat established that the set of non-constant solutions of $x\cdot
x^{\prime\prime}=x^{\prime}$ is strongly minimal (see [17] for an
explanation). Poizat’s arguments were generalized by Brestovski [3] to a class
of very specifically chosen order two differential equations with constant
coefficients.

  2. (2)

Hrushovski’s work [12] around the Mordell-Lang conjecture proved the strong
minimality of Manin kernels of nonisotrivial simple abelian varieties. It uses
specific properties of abelian varieties as well as model-theoretic techniques
around _modularity_ of strongly minimal sets.

  3. (3)

Nagloo and Pillay [20] show that results of the Japanese school of
differential algebra [18, 22, 23, 24, 25, 33, 34, 35, 36] imply that Painlevé
equations with generic coefficients are strongly minimal. The techniques
employed are differential algebraic and valuation theoretic, relying on very
specific properties of the equations.

  4. (4)

Work of Freitag and Scanlon [11] for the differential equation satisfied by
the $j$-function ultimately relies on point-counting and o-minimality via the
Pila-Wilkie theorem as applied in [27, 28]; the argument there is very
specific to the third order nonlinear differential equation satisfied by the
$j$-function. Later, Aslanyan [1] produced another proof, ultimately relying
on similar (stronger) inputs of [29].

  5. (5)

Casale, Freitag and Nagloo [4] show that equations satisfied by
$\Gamma$-automorphic functions on the upper half-plane for $\Gamma$ a Fuchsian
group of the first kind are strongly minimal. The arguments use differential
Galois theory with some additional analytic geometry, and the techniques again
are very specific to the third order equations of this specific form.

  6. (6)

Jaoui shows that generic planar vector fields over the constants give rise to
strongly minimal order two differential varieties [14]. The arguments rely on
various sophisticated analytic techniques and results from foliation theory,
some of which are particular to the specific class of equations considered.

  7. (7)

Blázquez-Sanz, Casale, Freitag, and Nagloo [2] prove the strong minimality of
certain general Schwarzian differential equations.

  8. (8)

Freitag, Jaoui, Marker, and Nagloo [7] show that various equations of Liénard-
type are strongly minimal using techniques from valuation theory.

We should also mention that strong minimality in this context was perhaps
first studied by Painlevé using different language in [26]. Painlevé
conjectured the strong minimality of various classes of differential
equations, where the notion is equivalent to _Umemura’s Condition (J)._ See
[20] for a discussion of these connections. We believe that the above list,
together with a specific example of [6] constitutes the entire list of
differential equations (of order at least two) which have been proven to be
strongly minimal. Most of the techniques in the above listed results apply
only to specific equations or narrow classes of equations and rely on specific
properties of those classes in proving strong minimality. Our goal in this
article will be to develop a rather more general approach which applies widely
to equations with at least one differentially transcendental coefficient.

###  1.1. Our approach and results

Let $f\in k\\{x\\}$. Generally speaking, when attempting to prove strong
minimality111Equivalently, there are no infinite differential subvarieties. of
some differential variety

$V=Z(f)=\\{a\in\mathcal{U}\,|\,f(a)=0\\},$

there are two phenomena which make the task difficult:

  1. (1)

There is no a priori upper bound on the degree of the differential polynomials
which define a differential subvariety of $V$.

  2. (2)

The differential polynomials used to define a differential subvariety might
(necessarily) have coefficients from a differential field extension of the
field of $k$.

There are structure theorems related to (1) but only in special cases. See for
instance [9] when the subvarieties are co-order one in $V$. Controlling the
field extension in (2) is a key step in various recent works [11, 14, 20].
This is most often accomplished by noting that _stable embeddedness_ of the
generic type of $V$ implies that the generators of the field of definition of
a forking extension can be assumed to themselves realize the generic type of
$V$ – see explanations in [4, 11]. In recent work, Freitag and Moosa [10]
introduce a new invariant of a type, which more closely controls the structure
over which the forking extension of a type is defined:

######  Definition 1.1.

Suppose $p\in S(A)$ is a stationary type of $U$-rank $>1$. By the degree of
nonminimality of $p$, denoted by $\operatorname{nmdeg}(p)$, we mean the least
positive integer $k$ such that for a Morley sequence of $p$ of length $k$, say
$(a_{1},\dots,a_{k})$, $p$ has a nonalgebraic forking extension over
$A,a_{1},\dots,a_{k}$. If $\operatorname{RU}(p)\leq 1$ then we set
$\operatorname{nmdeg}(p)=0$.

In the theory of differentially closed fields of characteristic zero, Freitag
and Moosa [10] give an upper bound for the degree of nonminimality in terms of
Lascar rank:

######  Theorem 1.2.

Let $p\in S(k)$ have finite rank. Then
$\operatorname{nmdeg}(p)\leq\operatorname{RU}(p)+1.$

Let $a\models p$, we will call the transcendence degree of the differential
field $k\langle a\rangle/k$ the order of $p$. When $p$ is the generic type of
a differential variety $V$, we also call this the order222These notions of
order agree with the previously mentioned definition of order for differential
polynomials. of $V$. The order of $p$ is an upper bound for the Morley rank of
$p$. The Morley rank of $p$ is a bound for the Lascar rank of $p$. For proofs
of these facts, see [17]. It follows that _if_ the type $p$ of a generic
solution of an order $n$ differential equation over $k$ has a nonalgebraic
forking extension over some differential field extension, then already $p$ has
such a forking extension over $k\langle a_{1},\ldots,a_{n+1}\rangle$ where the
$a_{i}$ are from a Morley sequence in the type of $p$ over $k$. This
consequence of Theorem 1.2 will be essential to our approach to handling issue
(2) above.

Our approach to issue (1) follows a familiar general strategy of reducing
certain problems for nonlinear differential equations to related problems for
associated linear differential equations. For instance, [31] applies a
strategy of this nature to establish results around the Zilber trichotomy,
while [4, 19] use this strategy to establish irreducibility of solutions to
automorphic and Painlevé equations using certain associated Riccati equations.
Our technique fits into this general framework and relies on Kolchin’s
differential tangent space, which will provide the linear equations associated
with the original nonlinear differential variety $V$. Our approach to the
associated linear equations has been under development in the thesis of Wolf
[37] and the forthcoming thesis of DeVilbiss which gives an approach to
calculating the Lascar rank of underdetermined systems of linear differential
equations.

When we call a differential polynomial $f(x)$ _generic_ of order $h$ and
degree $d$, if $f$ is a linear combination of all monomials of degree no more
than $d$ on the variables $x,x^{\prime},\ldots,x^{(h)}$ with independent
differentially transcendental coefficients.

Our main theorem is:

######  Theorem 1.3.

Let $f(x)$ be a generic differential polynomial of order $h>1$ and degree $d$.
Let $p$ be the type of a generic solution to $Z(f)$. If $d\geq
2\cdot(\operatorname{nmdeg}(p)+1),$ then $Z(f)$ is strongly minimal.

Since
$\operatorname{nmdeg}(p)\leq\operatorname{RU}(p)+1\leq\operatorname{ord}(f)+1$,
the following corollary is immediate:

######  Corollary 1.4.

Let $f(x)$ be a generic differential polynomial of order $h>1$ and degree $d$.
If $d\geq 2\cdot(h+2),$ then $Z(f)$ is strongly minimal.

This answers Question 7 of [32] for sufficiently large degree, any order, and
nonconstant coefficients. As described above, Jaoui [14] has recently answered
the order two case of Question 7 of [32] for constant coefficients. We
conjecture a more general form of Theorem 1.3:

######  Conjecture 1.5.

Generic differential equations of fixed order and degree greater than one are
strongly minimal.

To the non-model theorist, it does not seem obvious why strong minimality
plays a central role in the theory of algebraic differential equations, but
there seem to be two important factors behind this:

  * •

Once strong minimality of an equation is established, powerful results having
their origins in geometric stability theory can be employed (e.g. the Zilber
trichotomy, discussed next).

  * •

Among nonlinear equations, the results of this paper and those of [14] show
that strong minimality holds ubiquitously.

Even when an equation is not strongly minimal, it is often true that questions
about the solutions can be reduced to questions about solutions of certain
associated minimal equations coming from a notion called _semi-minimal
analysis_ ; for instance, see [8]. We now state the Zilber trichotomy for
strongly minimal sets adapted to the setting of differentially closed fields:

######  Fact 1.6 ([13],[31]).

Let $X$ be a strongly minimal set. Then exactly one of the following holds:

  1. (1)

(non-locally modular) $X$ is nonorthogonal to $\mathbb{C}$,

  2. (2)

(locally modular, nontrivial) $X$ is nonorthogonal to the (unique) smallest
Zariski-dense definable subgroup of a simple abelian variety $A$ which does
not descend to $\mathbb{C}$,

  3. (3)

(trivial) $X$ is geometrically trivial.

Nonorthogonality is a natural equivalence relation on definable sets. In the
context of strongly minimal sets, $X$ is nonorthogonal to $Y$ if and only if
there is a generic finite-to-finite definable correspondence between $X$ and
$Y$. A strongly minimal set is geometrically trivial if whenever
$a_{1},\ldots,a_{n}\in X$ are are dependent in the sense of forking, there is
a pair of $a_{i},a_{j}$ which is dependent. In terms of algebraic relations in
our setting, anytime $a_{1},\ldots,a_{n}\in X$ are generic solutions of the
differential equation $X$, if

$\operatorname{trdeg}_{K}(K\langle a_{1},\ldots a_{n}\rangle)\leq
n\operatorname{ord}(X),$

then there is a pair such that

$\operatorname{trdeg}_{K}(K\langle a_{i},a_{j}\rangle)\leq
2\operatorname{ord}(X).$

While Fact 1.6 gives a rather complete classification of strongly minimal sets
in the nontrivial cases of the Zilber trichotomy, there is no general
classification of the geometrically trivial strongly minimal sets in
differentially closed fields.

We conjecture that generic differential equations are _geometrically trivial_
in a strong form:

######  Conjecture 1.7.

Any two solutions of a generic differential equation of fixed order and degree
greater than one are (differentially) algebraically independent.

In this paper, our techniques are applied to equations with differentially
transcendental coefficients, but this is not an inherent restriction of the
methods. For instance, in forthcoming work using these techniques joint with
Casale and Nagloo, we give a fundamentally new proof of the main theorem of
[11], proving that the equation satisfied by the j-function is strongly
minimal. See Section 5 for additional discussion.

###  1.2. Organization

In section 2, we set up the notation and background results we require.
Section 3 gives a new sufficient condition for the strong minimality of a
differential variety. Section 4 applies this condition to show that generic
differential equations are strongly minimal. Section 5 shows how one can
establish a weaker condition than strong minimality in a more computationally
straightforward manner and gives some open problems.

##  2\. Notation

Let $\mathcal{U}$ be a countably saturated differentially closed field of
characteristic zero. All of the fields we consider will be subfields of
$\mathcal{U}$. An affine _differential variety_ is the zero set of a (finite)
system of differential polynomial equations over (a finitely generated
subfield of) $\mathcal{U}$.

Let $(y_{1},\ldots y_{n})$ be a finite set of differential indeterminants over
$\mathcal{U}$ and let $\Theta$ denote the set of derivative operators on
$\mathcal{U}$. Since we are interested in differential fields with a single
derivation, $\Theta=\\{\delta^{k}:k\geq 0\\}$. A _ranking_ on $(y_{1},\ldots
y_{n})$ is a total ordering on the derivatives $\\{\theta
y_{j}:\theta\in\Theta,1\leq j\leq n\\}$ such that for all such derivatives
$u$, $v$, and all $\theta\in\Theta$, we have

$u\leq\theta u,\quad u\leq v\Rightarrow\theta u\leq\theta v.$

A ranking is _orderly_ if whenever the order of $\theta_{1}$ is lower than the
order of $\theta_{2}$, we have $\theta_{1}y_{i}<\theta_{2}y_{j}$ for any
$i,j$. An _elimination ranking_ is a ranking in which $y_{i}<y_{j}$ implies
$\theta_{1}y_{i}<\theta_{2}y_{j}$ for any $\theta_{1},\theta_{2}\in\Theta$.
For a $\delta$-polynomial $f(y_{1},\ldots,y_{n})$, the highest ranking $\theta
y_{j}$ appearing in $f$ is the _leader_ of $f$, denoted $u_{f}$. If $u_{f}$
has degree $d$ in $f$, we can rewrite $f$ as a polynomial in $u_{f}$,
$f=\sum_{i=0}^{d}I_{i}u_{f}^{i}$, where the _initial_ of $f$, $I_{d}$, is not
zero. The _separant_ of $f$ is the formal derivative $\frac{\partial
f}{\partial u_{f}}$. A detailed treatment of these definitions can be found in
[15, pg 75].

Let $\bar{a}\in\mathcal{U}$, and $F$ a differential subfield of $\mathcal{U}$.
There is a numerical polynomial $\omega(\bar{a}/F)$ called the _Kolchin
polynomial_ of $\bar{a}$ over $F$ such that for sufficiently large
$s\in\mathbb{N}$,

$\omega_{\bar{a}/F}(s)=\operatorname{trdeg}\left(F\left(\bar{a},\delta(\bar{a}),\ldots,\delta^{s}(\bar{a})\right)/F\right)$

(see [15, Theorem 6, pg 115]). When $X$ is a differential variety, that is, a
closed irreducible set in the Kolchin topology,
$\omega(X/F):=\omega(\bar{a}/F)$ where $\bar{a}$ is a generic point on $X$
over $F$. Since we are only concerned with differential fields with a single
derivation, the degree of $\omega(X/F)$, sometimes called the _differential
type_ is either 0 or 1. The leading coefficient of $\omega(X/F)$ is called the
_typical differential dimension_. Crucially for our purposes, the Kolchin
polynomial witnesses forking in the sense of model theory, i.e.,

$\bar{a}\mathop{\mathchoice{\displaystyle\kern 5.71527pt\hbox
to0.0pt{\hss$\displaystyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\displaystyle\smile$\hss}\kern 5.71527pt}{\textstyle\kern
5.71527pt\hbox to0.0pt{\hss$\textstyle\mid$\hss}\lower 3.87495pt\hbox
to0.0pt{\hss$\textstyle\smile$\hss}\kern 5.71527pt}{\scriptstyle\kern
2.80048pt\hbox to0.0pt{\hss$\scriptstyle\mid$\hss}\lower 1.89871pt\hbox
to0.0pt{\hss$\scriptstyle\smile$\hss}\kern 2.80048pt}{\scriptscriptstyle\kern
1.42882pt\hbox to0.0pt{\hss$\scriptscriptstyle\mid$\hss}\lower 0.96873pt\hbox
to0.0pt{\hss$\scriptscriptstyle\smile$\hss}\kern
1.42882pt}}_{A}C\Longleftrightarrow\omega_{\bar{a}/A}(s)=\omega_{\bar{a}/A\cup
C}(s).$

Let $X\subset\mathcal{U}^{n}$ be a differential variety and let $I(X)$ be the
ideal of differential polynomials vanishing on $X$. The _differential tangent
bundle_ $T\Delta X$ is the differential variety given by

$\left\\{(x_{1},\ldots,x_{n},w_{1},\ldots,w_{n})\in\mathcal{U}^{2n}:\forall
f\in I(X),f(\bar{x})=0,\sum_{\begin{subarray}{c}i\leq n\\\
j\leq\operatorname{ord}(f)\end{subarray}}\frac{\partial f}{\partial
x_{i}^{(j)}}(\bar{x})w_{i}^{(j)}=0\right\\}$

where $\frac{\partial f}{\partial x_{i}^{(j)}}(\bar{x})$ is a formal partial
derivative where $x_{i}^{(j)}$ are considered as algebraic variables. Given
$\bar{a}\in X$, the _differential tangent space_ of $X$ over $\bar{a}$,
denoted $T_{\bar{a}}^{\Delta}X$ is the fiber of $T^{\Delta}X$ over $\bar{a}$.
The following lemma is a corollary of [16, Theorem 1, pg 199].

######  Lemma 2.1.

Let $F$ be a differential field, $X$ a differential variety defined over $F$,
and a point $a\in X$ outside of the vanishing locus of the separant. Then
$\omega\left(X/F\right)=\omega\left(T^{\Delta}_{a}(X)/F\langle
a\rangle\right)$.

##  3\. A general sufficient criterion for strong minimality

Let $f(x)$ be an order $n\geq 1$ non-linear differential polynomial in one
variable without a constant term. Let $\bar{\alpha}$ denote the coefficients
of $f$ and let $\alpha_{0}$ be differentially transcendental over
$\bar{\alpha}$. Let $V_{0}$ be the differential variety corresponding to
$f(x)=\alpha_{0}$. Our goal in this section is to find sufficient conditions
under which such a variety $V_{0}$ is strongly minimal.

Our next proposition shows that when $\alpha_{0}$ is differentially
transcendental over $\mathbb{Q}\langle\bar{\alpha}\rangle$, there are no
proper subvarieties of $V_{0}$ which are defined over the field
$\mathbb{Q}\langle\bar{\alpha},\alpha_{0}\rangle.$ Though the argument is
simple, an elaboration of the technique in the proof will be used in the more
difficult general case where one extends the field of coefficients.

######  Proposition 3.1.

Let $f(x)$ be a non-linear order $h\geq 1$ differential polynomial with
coefficients $\bar{\alpha}$, let $\alpha_{0}$ be differentially transcendental
over $\mathbb{Q}\langle\bar{\alpha}\rangle$, and let $V_{0}$ be the
differential variety defined by $f(x)=\alpha_{0}$. Then $V_{0}$ has no
infinite subvarieties that are defined over
$\mathbb{Q}\langle\bar{\alpha},\alpha_{0}\rangle$.

###### Proof.

Suppose towards a contradiction that $W_{0}$ is an infinite proper subvariety
of $V_{0}$ defined over $\mathbb{Q}\langle\bar{\alpha},\alpha_{0}\rangle$.
Then $W_{0}$ is given by some positive order $\delta$-polynomial
$g(x)\in\mathbb{Q}\langle\bar{\alpha},\alpha_{0}\rangle\\{x\\}$. By clearing
the denominators of $\alpha_{0}$, we can write
$g(x,\alpha_{0})\in\mathbb{Q}\langle\bar{\alpha}\rangle\\{x,\alpha_{0}\\}$.
For ease of notation, let $k=\mathbb{Q}\langle\bar{\alpha}\rangle$.

Let $V$ be the $\delta$-variety given by $f(x)=y$ and let $W$ be given by
$g(x,y)=0$ so that each instance of $\alpha_{0}$ is replaced with the variable
$y$. These varieties are now defined by $\delta$-polynomials in two variables
with coefficients in $k$ and $W\subsetneq V$. Let $a=(a_{1},a_{2})$ be a
generic point of $W$ over $k$. Since $\alpha_{0}$ is differentially
transcendental over $\bar{\alpha}$ the locus of $y$ over $k$ is
$\mathbb{A}^{1}$, so it follows that $W$ is an infinite rank (proper)
subvariety of $V$. Consider the orderly ranking with $x$ ranked higher than
$y$.

We claim that the generic point $a$ of $W$ lies outside the locus on $V$ where
the separant of $f(x)-y$ vanishes (we will call this the singular locus of
$V$). This follows because the locus of the separant of $f$ inside of $V$ is
finite rank (to see this, note that the separant is a differential polynomial
in $k\\{x\\}$ so its generic solution has $x$-coordinate differentially
algebraic over $k$). From the fact that $a$ lies outside the singular locus of
$V$ and the singular locus of $W$ (since $a$ is generic on $W$), it follows
that the Kolchin polynomials of $T^{\Delta}_{a}W$ and $T^{\Delta}_{a}V$ are
equal to the Kolchin polynomials of $W$ and $V$, respectively, and so
$T^{\Delta}_{a}W\subsetneq T^{\Delta}_{a}V$.

For $0\leq i\leq n$, let

$\beta_{i}(x)=\frac{\partial f}{\partial x^{(i)}}(x)$

denote the formal derivative of $f$ with respect to the $i$th derivative of
$x$. Using this notation, the differential tangent space $T^{\Delta}_{a}V$ is
the set of $(w,z)$ satisfying the linear differential equation

$z=\sum_{i=0}^{n}\beta_{i}(a)w^{(i)}.$

From this equation, we can see that $z$ is determined by our choice of $w$,
but $w$ may be chosen freely. This gives a definable bijection between
$T^{\Delta}_{a}V$ and $\mathbb{A}^{1}(\mathcal{U})$. Further, it follows that
$T^{\Delta}_{a}V$ has no infinite rank subspaces over $k\langle a\rangle$,
since if it did, we could consider the image of this subvariety under the
definable bijection to $\mathbb{A}^{1}(\mathcal{U})$. However,
$\mathbb{A}^{1}(\mathcal{U})$ has no infinite rank subsets, so the image must
have finite rank. Therefore, $\omega\left(T^{\Delta}_{a}W/k\langle
a\rangle\right)$ is finite, a contradiction. ∎

######  Remark 3.2.

Consider the statements:

  1. (1)

There is a definable bijection between the differential tangent space
$T^{\Delta}_{\bar{a}}V$ and $\mathbb{A}^{1}(\mathcal{U})$.

  2. (2)

The Lascar rank $\operatorname{RU}\left(T^{\Delta}_{\bar{a}}V\right)=\omega$.

  3. (3)

The differential tangent space $T^{\Delta}_{\bar{a}}V$ has no definable
infinite rank subspaces over $\mathbb{Q}\langle\bar{a},\bar{\alpha}\rangle$.

The previous argument shows that $(1)\Rightarrow(2)\Rightarrow(3)$ and this
chain of implications will be used throughout this paper. Although Condition
(3) is a priori the weakest of the three, it will be the one most often used.

######  Remark 3.3.

The previous result shows that under very general circumstances, for instance
when any single coefficient is differentially transcendental over the others,
the equation has no subvarieties over the coefficients of the equation itself.
We state the following result, but omit its proof, as it is analogous to the
previous proof and will not be used later in this paper.

######  Proposition 3.4.

Let $f$ be a differential polynomial in one variable and $V$ the zero set of
$f$. Let $\bar{a}$ denote the tuple of coefficients in $f$. If $\bar{a}$ has
some element $a_{1}$ such that $a_{1}$ is differentially transcendental over
$\mathbb{Q}\langle\bar{a}_{-1}\rangle$,333By $\bar{a}_{-1}$, we mean the tuple
$\bar{a}$ excluding $a_{1}$. then $V$ has no differential subvarieties over
$\mathbb{Q}\langle\bar{a}\rangle$ except perhaps the zero set given by the
monomial of which $a_{1}$ is a coefficient.

The previous proposition works in such generality, in part because we have
restricted the coefficient field. In various situations, identifying
differential subvarieties defined over the field of definition of a variety
$V$ is a _much easier problem_ than identifying differential subvarieties of
$V$ defined over differential field extensions. For instance, in [21],
Nishioka shows that the equations corresponding to automorphic functions of
dense subgroups of $SL_{2}$ have to differential subvarieties over
$\mathbb{C}$. In the special case of genus zero Fuchsian functions, a much
more difficult argument was required to extend the result to differential
subvarieties over differential field extensions [4], answering a long-standing
open problem of Painlevé.

There is one general purpose model theoretic tool which restricts the field
extensions one needs to consider. We will use a principle in stability theory,
generally related to _stable embeddedness_ (see for instance see [5] where
this general type of result is referred to as the _Shelah reflection
principle_). For the following result see Lemma 2.28 [30]:

######  Lemma 3.5.

In a stable theory, let $A\subseteq B$ and $p\in S(B)$ which forks over $A$.
Then there is an indiscernible sequence $(a_{i}:i\in\mathbb{N})$ such that the
canonical base of $p$ is contained in the definable closure of
$A,a_{1},\ldots,a_{d}$ where $a_{1},\ldots,a_{d}$ is a finite initial segment
of this indiscernible sequence.

Let $f$ and $V_{0}$ be as before and let $d\in\mathbb{N}$. Consider
$V_{0}^{[d]}$, the set of $d$-tuples so that each coordinate $x_{i}$ satisfies
$f(x_{i})=\alpha_{0}$. As before, we can replace each instance of $\alpha_{0}$
with a new variable $y$, resulting in a differential variety $V^{[d]}$ defined
by the system of equations:

$\left\\{\begin{array}[]{lcl}f(x_{1})&=&y\\\ f(x_{2})&=&y\\\ &\vdots&\\\
f(x_{d})&=&y\end{array}\right.$

######  Proposition 3.6.

Let $f(x)$ be a non-linear order $h\geq 1$ differential polynomial with
coefficients $\bar{\alpha}$, let $\alpha_{0}$ be differentially transcendental
over $\mathbb{Q}\langle\bar{\alpha}\rangle$, and let $V_{0}$ be the
differential variety defined by $f(x)=\alpha_{0}$. Suppose that for all
$d\in\mathbb{N}$ and indiscernible sequences $\bar{a}$ in the generic type of
$V_{0}$, the differential tangent space
$T^{\Delta}_{\bar{a}}\left(V^{[d]}\right)$ has no definable proper infinite
rank subspaces over $\mathbb{Q}\langle\bar{\alpha},\bar{a}\rangle$. Then
$V_{0}$ is strongly minimal.

###### Proof.

Suppose $V_{0}$ is not strongly minimal and let $p(x)\in
S_{1}\left(\mathbb{Q}\langle\bar{\alpha},\alpha_{0}\rangle\right)$ be the type
of a generic solution of $V_{0}$. By Proposition 3.1, $V_{0}$ does not have
any infinite subvarieties defined over
$\mathbb{Q}\langle\bar{\alpha},\alpha_{0}\rangle$, so $p$ has a forking
extension $q$ over a differential field extension
$K>\mathbb{Q}\langle\bar{\alpha},\alpha_{0}\rangle$. By Lemma 3.5, there is
some finite $d$ and a Morley sequence $(a_{1},\ldots,a_{d})$ for $q$ such that
$(a_{1},\ldots,a_{d})$ is not
$\mathbb{Q}\langle\bar{\alpha},\alpha_{0}\rangle$-independent. Consider the
minimal such $d$. Then
$\operatorname{tp}\left(a_{1}/\mathbb{Q}\langle\bar{\alpha},\alpha_{0},a_{2},a_{3},\ldots,a_{d}\rangle\right)$
forks over $\mathbb{Q}\langle\bar{\alpha},\alpha_{0}\rangle$. Since these are
types over differential fields, this happens exactly when the Kolchin
polynomial of
$\left(a_{1}/\mathbb{Q}\langle\bar{\alpha},\alpha_{0},a_{2},a_{3},\ldots,a_{d}\rangle\right)$
is strictly less than the Kolchin polynomial of
$\left(a_{1}/\mathbb{Q}\langle\bar{\alpha},\alpha_{0}\rangle\right)$ because
the Kolchin polynomial witnesses forking.

Thus, there is a differential polynomial
$g(x)\in\mathbb{Q}\langle\bar{\alpha},\alpha_{0},a_{2},\ldots,a_{d}\rangle\\{x\\}$
so that $g(a_{1})=0$ and $g$ has order strictly less than the order of $f$. By
clearing denominators, we can write
$g(x_{1},\ldots,x_{d})\in\mathbb{Q}\langle\bar{\alpha},\alpha_{0}\rangle\\{x_{1},\ldots,x_{d}\\}$
such that $g(a_{1},\ldots,a_{d})=0$. Let $U_{0}\subset V_{0}^{[d]}$ be the
vanishing set of $g(x_{1},\ldots,x_{d})$. Just as with $V$, we can replace
$\alpha_{0}$ with a new variable $y$ after clearing denominators again, giving
a $\mathbb{Q}\langle\bar{\alpha}\rangle$-polynomial $g(x_{1},\ldots,x_{d},y)$
and the corresponding variety $U\subset V^{[d]}$. The Kolchin polynomial
$\omega\left(U/\mathbb{Q}\langle\bar{\alpha}\rangle\right)$ is nonconstant.
Let $\bar{a}=(a_{1},\ldots,a_{d},\alpha_{0})$ and notice that $\bar{a}$ is a
generic point of $U$ over $\mathbb{Q}\langle\alpha\rangle$. By Lemma 2.1, the
Kolchin polynomial of the differential tangent space
$\omega\left(T^{\Delta}_{\bar{a}}(U)/\mathbb{Q}\langle\bar{\alpha},\bar{a}\rangle\right)$
is also nonconstant, so $T^{\Delta}_{\bar{a}}(V^{[d]})$ has an infinite rank
subspace over $\mathbb{Q}\langle\bar{\alpha,}\bar{a}\rangle$, a contradiction
to our assumption. ∎

######  Remark 3.7.

Using Lemma 3.5 together with Proposition 3.6 gives a strategy for
establishing the strong minimality of nonlinear differential equations with
generic coefficients, but _only if one can verify the hypothesis of
Proposition 3.6_. A priori, this looks quite hard since it would require the
analysis of systems of linear differential equations in $n$ variables for all
$n\in\mathbb{N}.$ This may be possible via a clever inductive argument for
specially selected classes of equations, but Theorem 1.2 gives a bound for the
number of variables we need to consider.

######  Theorem 3.8.

Let $f(x)$ be a non-linear order $h\geq 1$ differential polynomial with
coefficients $\bar{\alpha}$, let $\alpha_{0}$ be differentially transcendental
over $\mathbb{Q}\langle\bar{\alpha}\rangle$, and let $V_{0}$ be the
differential variety defined by $f(x)=\alpha_{0}$. Let $p$ be the generic type
of $V_{0}$. Suppose that for
$d\leq\operatorname{nmdeg}{p}+1\leq\text{ord}(V_{0})+2$ and any indiscernible
sequence $\bar{a}=(a_{1},\ldots,a_{d})$ in the generic type of $V_{0}$ with
$a_{1},\ldots,a_{d}$ algebraically independent, the differential tangent space
$T^{\Delta}_{\bar{a}}\left(V^{[d]}\right)$ has no definable proper infinite
rank subspaces over $\mathbb{Q}\langle\bar{\alpha},\bar{a}\rangle$. Then
$V_{0}$ is strongly minimal.

###### Proof.

By Proposition 3.1, there are no subvarieties of $V_{0}$ defined over the
differential field generated by the coefficients of $f$. So, we need only
consider forking extensions of the generic type of $V_{0}$. By Theorem 1.2, if
there is an infinite proper differential subvariety of $V_{0}$, then it is
defined over (the algebraic closure of) a Morley sequence of length at most
$\operatorname{nmdeg}(p)$ which is at most $h+1$. Thus, there is a proper
subvariety of $W_{0}\subset V_{0}^{[d]}$ which surjects onto the first $d-1$
coordinates such that the fiber over a generic point in the first $d-1$
coordinates is a forking extension of the generic type of $V_{0}$. But then by
the argument of Proposition 3.6, there is a definable proper infinite rank
subspace of $T^{\Delta}_{\bar{a}}\left(V^{[d]}\right)$ over
$\mathbb{Q}\langle\bar{\alpha},\bar{a}\rangle$. ∎

##  4\. Strong minimality of generic equations

###  4.1. A first example

######  Theorem 4.1.

Let $X_{\alpha}$ be the differential variety given by

(1) $x^{\prime\prime}+\sum_{i=1}^{n}\alpha_{i}x^{i}=\alpha$

for some $n\geq 8$, where $(\alpha,\alpha_{0},\ldots,\alpha_{n})$ is a tuple
of independent differential transcendentals over $\mathbb{Q}.$ Then
$X_{\alpha}$ is strongly minimal.

###### Proof.

To show that equation 1 is strongly minimal, by the explanation following
Theorem 1.2, we need only show that given any solutions
$a_{1},a_{2},a_{3},a_{4}$ to equation 1 that are algebraically independent
over $\mathbb{Q}\langle\alpha,\alpha_{1},\ldots,\alpha_{n}\rangle$,444That is,
$a_{1},a_{2},a_{3},a_{4}$ satisfy no polynomial relation over
$\mathbb{Q}\langle\alpha,\alpha_{1},\ldots,\alpha_{n}\rangle$. we cannot have
that the transcendence degree of $\mathbb{Q}\langle
a_{1},\alpha,\alpha_{1},\ldots,\alpha_{n},a_{2},\ldots,a_{4}\rangle$ over
$\mathbb{Q}\langle\alpha,\alpha_{1},\ldots,\alpha_{n},a_{2},\ldots,a_{4}\rangle$
is one.

Observe that the differential tangent space
$T_{\bar{a}}^{\Delta}\left(X^{[4]}\right)$ after eliminating $y$ is given by
the system:

$\left\\{\begin{array}[]{lcl}u_{0}^{\prime\prime}+\left(\sum_{i=1}^{n}ia_{1}^{i-1}\alpha_{i}\right)u_{0}&=&v_{0}^{\prime\prime}+\left(\sum_{i=1}^{n}ia_{2}^{i-1}\alpha_{i}\right)v_{0}\\\
u_{0}^{\prime\prime}+\left(\sum_{i=1}^{n}ia_{1}^{i-1}\alpha_{i}\right)u_{0}&=&w_{0}^{\prime\prime}+\left(\sum_{i=1}^{n}ia_{3}^{i-1}\alpha_{i}\right)w_{0}\\\
u_{0}^{\prime\prime}+\left(\sum_{i=1}^{n}ia_{1}^{i-1}\alpha_{i}\right)u_{0}&=&z_{0}^{\prime\prime}+\left(\sum_{i=1}^{n}ia_{4}^{i-1}\alpha_{i}\right)z_{0}\\\
\end{array}\right.$

For $j=1,\ldots,4$, we let $\beta_{j}=\sum_{i=0}^{n}ia_{j}^{i-1}\alpha_{i}$.
We argue that $\beta_{1},\ldots,\beta_{4}$ are independent differential
transcendentals. Note that

$\left(\begin{matrix}na_{1}^{n-1}&(n-1)a_{1}^{n-2}&\ldots&2a_{1}&1\\\
na_{2}^{n-1}&(n-1)a_{2}^{n-2}&\ldots&2a_{2}&1\\\
na_{3}^{n-1}&(n-1)a_{3}^{n-2}&\ldots&2a_{3}&1\\\
na_{4}^{n-1}&(n-1)a_{4}^{n-2}&\ldots&2a_{4}&1\\\
\end{matrix}\right)\left(\begin{matrix}\alpha_{n}\\\ \alpha_{n-1}\\\ \vdots\\\
\alpha_{1}\end{matrix}\right)=\left(\begin{matrix}\beta_{1}\\\ \beta_{2}\\\
\beta_{3}\\\ \beta_{4}\end{matrix}\right)$

We claim that any four columns of the matrix of $a_{i}$’s are linearly
independent. To see this, note that if not then the vanishing of the
corresponding determinant shows that there is a nontrivial polynomial relation
which holds of $a_{1},\ldots,a_{4}.$ This contradicts the fact that $a_{1}$
satisfies an order one equation over $a_{2},\ldots,a_{4}.$ By the independence
of $\alpha_{1},\ldots,\alpha_{n}$ there are at least four of the $\alpha_{i}$
which are independent differential transcendentals over the other $\alpha_{i}$
and $a_{1},\ldots,a_{4}.$ Without loss of generality, assume
$\alpha_{1},\ldots,\alpha_{4}$ are independent differential transcendentals
over
$\mathbb{Q}\langle\alpha_{5},\ldots,\alpha_{n},a_{1},\ldots,a_{4}\rangle$.
Then since the last four columns of the above matrix of $a_{i}$ are linearly
independent, it follows that $\alpha_{1},\ldots,\alpha_{4}$ are interalgebraic
with $\beta_{1},\ldots,\beta_{4}$ over $\mathbb{Q}\langle
a_{1},\ldots,a_{4},\alpha_{5},\ldots,\alpha_{n}\rangle.$ It follows that
$\beta_{1},\ldots,\beta_{4}$ are independent differential transcendentals over
$\mathbb{Q}\langle a_{1},\ldots,a_{4},\alpha_{5},\ldots,\alpha_{n}\rangle$.

######  Lemma 4.2.

A linear system of the form

(2)
$\left\\{\begin{array}[]{lcl}u_{0}^{\prime\prime}+\beta_{1}u_{0}&=&v_{0}^{\prime\prime}+\beta_{2}v_{0}\\\
u_{0}^{\prime\prime}+\beta_{1}u_{0}&=&w_{0}^{\prime\prime}+\beta_{3}w_{0}\\\
u_{0}^{\prime\prime}+\beta_{1}u_{0}&=&z_{0}^{\prime\prime}+\beta_{4}z_{0}\\\
\end{array}\right.$

with $\beta_{1},\ldots,\beta_{4}$ independent differential transcendentals has
no infinite rank subvarieties.

###### Proof.

We will prove that this system has no infinite rank subvarieties by proving
that the solution set is in definable bijection with
$\mathbb{A}^{1}(\mathcal{U})$. This is constructed by composing a series of
linear substitutions.

First, we substitute $u_{1}$ for $u_{0}$ where $u_{0}=u_{1}+v_{0}$. This
reduces the order of $v_{0}$ in the top equation, resulting in the system

$\left\\{\begin{array}[]{lcl}u_{1}^{\prime\prime}+\beta_{1}u_{1}&=&(\beta_{2}-\beta_{1})v_{0}\\\
u_{1}^{\prime\prime}+\beta_{1}u_{1}+v_{0}^{\prime\prime}+\beta_{1}v_{0}&=&w_{0}^{\prime\prime}+\beta_{3}w_{0}\\\
u_{1}^{\prime\prime}+\beta_{1}u_{1}+v_{0}^{\prime\prime}+\beta_{1}v_{0}&=&z_{0}^{\prime\prime}+\beta_{4}z_{0}\\\
\end{array}\right.$

To reduce the order $v_{0}$ in the lower equations we substitute $w_{1},z_{1}$
for $w_{0},z_{0}$ where $w_{0}=w_{1}+v_{0},z_{0}=z_{1}+v_{0}$. Then we have

$\left\\{\begin{array}[]{lcl}u_{1}^{\prime\prime}+\beta_{1}u_{1}&=&(\beta_{2}-\beta_{1})v_{0}\\\
u_{1}^{\prime\prime}+\beta_{1}u_{1}+(\beta_{1}-\beta_{3})v_{0}&=&w_{1}^{\prime\prime}+\beta_{3}w_{1}\\\
u_{1}^{\prime\prime}+\beta_{1}u_{1}+(\beta_{1}-\beta_{4})v_{0}&=&z_{1}^{\prime\prime}+\beta_{4}z_{1}\\\
\end{array}\right.$

Solving the top equation for $v_{0}$ in terms of $u_{1}$ and plugging this in
for $v_{0}$ allows us to eliminate $v_{0}$ from lower equations, resulting in
the system

$\left\\{\begin{array}[]{lcl}A_{2,0}u_{1}^{\prime\prime}+A_{0,0}u_{1}&=&w_{1}^{\prime\prime}+\beta_{3}w_{1}\\\
C_{2,0}u_{1}^{\prime\prime}+C_{0,0}u_{1}&=&z_{1}^{\prime\prime}+\beta_{4}z_{1}\end{array}\right.$

where (after some simplification)

$\displaystyle A_{2,0}$
$\displaystyle:=\frac{\beta_{2}-\beta_{3}}{\beta_{2}-\beta_{1}},\quad
A_{0,0}:=\beta_{1}A_{2,0}$ $\displaystyle C_{2,0}$
$\displaystyle:=\frac{\beta_{2}-\beta_{4}}{\beta_{2}-\beta_{1}},\quad
C_{0,0}:=\beta_{1}C_{2,0}.$

We again reduce the order of the variable in the top equation by substituting
$u_{2}$ for $u_{1}$ defined by $u_{1}=u_{2}+\frac{1}{A_{2,0}}w_{1}$ resulting
in the system

$\left\\{\begin{array}[]{lcl}A_{2,0}u_{2}^{\prime\prime}+A_{0,0}u_{2}&=&B_{1,1}w_{1}^{\prime}+B_{0,1}w_{1}\\\
C_{2,0}u_{2}^{\prime\prime}+C_{0,0}u_{2}+D_{2,1}w_{1}^{\prime\prime}+D_{1,1}w_{1}^{\prime}+D_{0,1}w_{1}&=&z_{1}^{\prime\prime}+\beta_{4}z_{1}\end{array}\right.$

where

$\begin{array}[]{lll}&B_{1,1}:=-2A_{2,0}\left(\frac{1}{A_{2,0}}\right)^{\prime},&\
B_{0,1}:=\beta_{3}-A_{2,0}\left(\frac{1}{A_{2,0}}\right)^{\prime\prime}-\frac{A_{0,0}}{A_{2,0}}\\\
D_{2,1}:=\frac{C_{2,0}}{A_{2,0}},&D_{1,1}:=2C_{2,0}\left(\frac{1}{A_{2,0}}\right)^{\prime},&D_{0,1}:=C_{2,0}\left(\frac{1}{A_{2,0}}\right)^{\prime\prime}+\frac{C_{0,0}}{A_{2,0}}.\end{array}$

We next reduce the order of $w_{1}$ in lower equations with the substitution
$z_{2}$ for $z_{1}$ defined by $z_{1}=z_{2}+D_{2,1}w_{1}$. Now we have the
system

$\left\\{\begin{array}[]{lcl}A_{2,0}u_{2}^{\prime\prime}+A_{0,0}u_{2}&=&B_{1,1}w_{1}^{\prime}+B_{0,1}w_{1}\\\
C_{2,0}u_{2}^{\prime\prime}+C_{0,0}u_{2}+E_{1,1}w_{1}^{\prime}+E_{0,1}w_{1}&=&z_{2}^{\prime\prime}+\beta_{4}z_{2}\end{array}\right.$

where

$E_{1,1}:=D_{1,1}-D_{2,1}^{\prime},\quad
E_{0,1}:=D_{0,1}-D_{2,1}^{\prime\prime}-\beta_{4}D_{2,1}.$

Next we reduce the order of $u_{2}$ in the top equation by substituting
$w_{2}$ for $w_{1}$ with $w_{1}=w_{2}+\frac{A_{2,0}}{B_{1,1}}u_{2}^{\prime}$
resulting in the system

$\left\\{\begin{array}[]{lcl}A_{1,1}u_{2}^{\prime}+A_{0,1}u_{2}&=&B_{1,1}w_{2}^{\prime}+B_{0,1}w_{2}\\\
C_{2,1}u_{2}^{\prime\prime}+C_{1,1}u_{2}^{\prime}+C_{0,1}u_{2}+E_{1,1}w_{2}^{\prime}+E_{0,1}w_{2}&=&z_{2}^{\prime\prime}+\beta_{4}z_{2}\\\
\end{array}\right.$

where

$\displaystyle A_{1,1}$
$\displaystyle:=-B_{1,1}\left(\frac{A_{2,0}}{B_{1,1}}\right)^{\prime}-B_{0,1}\frac{A_{2,0}}{B_{1,1}},\quad
A_{0,1}:=A_{0,0}$ $\displaystyle C_{2,1}$
$\displaystyle:=C_{2,0}+E_{1,1}\frac{A_{2,0}}{B_{1,1}},\
C_{1,1}:=E_{1,1}\left(\frac{A_{2,0}}{B_{1,1}}\right)^{\prime}+E_{0,1}\frac{A_{2,0}}{B_{1,1}},\quad
C_{0,1}:=C_{0,0}.$

The reduction in order of the top equation continues with the replacement of
$u_{2}$ with $u_{3}$ given by $u_{2}=u_{3}+\frac{B_{1,1}}{A_{1,1}}w_{2}$. This
results in the system

$\left\\{\begin{array}[]{lcl}A_{1,1}u_{3}^{\prime}+A_{0,1}u_{3}&=&B_{0,2}w_{2}\\\
C_{2,1}u_{3}^{\prime\prime}+C_{1,1}u_{3}^{\prime}+C_{0,1}u_{3}+D_{2,2}w_{2}^{\prime\prime}+D_{1,2}w_{2}^{\prime}+D_{0,2}w_{2}&=&z_{2}^{\prime\prime}+\beta_{4}z_{2}\end{array}\right.$

where

$\displaystyle B_{0,2}$
$\displaystyle:=B_{0,1}-A_{1,1}\left(\frac{B_{1,1}}{A_{1,1}}\right)^{\prime}-A_{0,1}\frac{B_{1,1}}{A_{1,1}}$
$\displaystyle D_{2,2}$ $\displaystyle:=C_{2,1}\frac{B_{1,1}}{A_{1,1}},\quad
D_{1,2}:=E_{1,1}+2C_{2,1}\left(\frac{B_{1,1}}{A_{1,1}}\right)^{\prime}+C_{1,1}\frac{B_{1,1}}{A_{1,1}},$
$\displaystyle D_{0,2}$
$\displaystyle:=E_{0,1}+C_{2,1}\left(\frac{B_{1,1}}{A_{1,1}}\right)^{\prime\prime}+C_{1,1}\left(\frac{B_{1,1}}{A_{1,1}}\right)^{\prime}+C_{0,1}\frac{B_{1,1}}{A_{1,1}}.$

Next we replace $z_{2}$ with $z_{3}$ given by $z_{2}=z_{3}+D_{2,2}w_{2}$ to
arrive at

$\left\\{\begin{array}[]{lcl}A_{1,1}u_{3}^{\prime}+A_{0,1}u_{3}&=&B_{0,2}w_{2}\\\
C_{2,1}u_{3}^{\prime\prime}+C_{1,1}u_{3}^{\prime}+C_{0,1}u_{3}+E_{1,2}w_{2}^{\prime}+E_{0,2}w_{2}&=&z_{3}^{\prime\prime}+\beta_{4}z_{3}\end{array}\right.$

where

$E_{1,2}:=D_{1,2}-D_{2,2}^{\prime},\quad
E_{0,2}:=D_{0,2}-D_{2,2}^{\prime\prime}-\beta_{4}D_{2,2}.$

Now we can solve the top equation for $w_{2}$ in terms of $u_{3}$ and plug the
resulting expression into the lower equations:

$C_{2,2}u_{3}^{\prime\prime}+C_{1,2}u_{3}^{\prime}+C_{0,2}u_{3}=z_{3}^{\prime\prime}+\beta_{4}z_{3}$

where

$\begin{array}[]{l}C_{2,2}:=C_{2,1}+E_{1,2}\frac{A_{1,1}}{B_{0,2}},\\\
C_{1,2}:=C_{1,1}+E_{1,2}\left(\frac{A_{1,1}}{B_{0,2}}\right)^{\prime}+E_{0,2}\frac{A_{1,1}}{B_{0,2}}+E_{1,2}\frac{A_{0,2}}{B_{0,2}},\\\
C_{0,2}:=C_{0,1}+E_{1,2}\left(\frac{A_{0,1}}{B_{0,2}}\right)^{\prime}+E_{0,2}\frac{A_{0,1}}{B_{0,2}}.\end{array}$

Next we perform analagous substitutions to eliminate $z_{3}$ from the top
equation, beginning with substituting $u_{4}$ for $u_{3}$ defined by
$u_{3}=u_{4}+\frac{1}{C_{2,2}}z_{3}$, so we have

$\left\\{\begin{array}[]{lcl}C_{2,2}u_{4}^{\prime\prime}+C_{1,2}u_{4}^{\prime}+C_{0,2}u_{4}&=&F_{1,1}z_{3}^{\prime}+F_{0,1}z_{3}\\\
\end{array}\right.$

where

$F_{1,1}:=-2C_{2,2}\left(\frac{1}{C_{2,2}}\right)^{\prime}-\frac{C_{1,2}}{C_{2,2}},\quad
F_{0,1}:=\beta_{4}-C_{2,2}\left(\frac{1}{C_{2,2}}\right)^{\prime\prime}-C_{1,2}\left(\frac{1}{C_{2,2}}\right)^{\prime}-\frac{C_{0,2}}{C_{2,2}}.$

Next, substitute $z_{4}$ for $z_{3}$ where
$z_{3}=z_{4}+\frac{C_{2,2}}{F_{1,1}}u_{4}^{\prime}$. This will result in the
equation

$C_{1,3}u_{4}^{\prime}+C_{0,3}u_{4}=F_{1,1}z_{4}^{\prime}+F_{0,1}z_{4}$

where

$C_{1,3}:=C_{1,2}-F_{1,1}\left(\frac{C_{2,2}}{F_{1,1}}\right)^{\prime}+F_{0,1}\frac{C_{2,2}}{F_{1,1}},\quad
C_{0,3}:=C_{0,2}.$

Replace $u_{4}$ with $u_{5}$ defined by
$u_{4}=u_{5}+\frac{F_{1,1}}{C_{1,3}}z_{4}$, giving us the equation

(3) $C_{1,3}u_{5}^{\prime}+C_{0,3}u_{5}=F_{0,2}z_{4}$

where

$F_{0,2}:=F_{0,1}-C_{1,3}\left(\frac{F_{1,1}}{C_{1,3}}\right)^{\prime}+F_{0,1}\frac{F_{1,1}}{C_{1,3}}.$

Any solution to Equation 3 is determined by the value of $u_{5}$, so the
solution set is in definable bijection with $\mathbb{A}^{1}(\mathcal{U})$.
Each of these linear substitutions gives rise to a definable bijection between
systems so long as the substitutions are well-defined, i.e., that the
denominators of the coefficients are all non-zero. For this procedure to give
a definable bijection from the original system 2 to
$\mathbb{A}^{1}(\mathcal{U})$, we must verify that the following expressions
are not zero:

$\beta_{2}-\beta_{1},A_{2,0},B_{1,1},A_{1,1},B_{0,2},C_{2,2},F_{1,1},C_{1,3},\text{
and }F_{0,2}.$

The expression $\beta_{2}-\beta_{1}$ is non-zero because
$\beta_{1},\beta_{2},\beta_{3},\beta_{4}$ are all distinct. Each of these
coefficients can be considered as a differential rational function in terms of
$\beta_{1},\beta_{2},\beta_{3},\beta_{4}$, and so they can be analyzed
according to a ranking on $\bar{\beta}$. We will show that these coefficients
are nonzero by showing that the initials of each are nonzero in some
elimination ranking. It then follows that the coefficients themselves are non-
zero because $\beta_{1},\beta_{2},\beta_{3},\beta_{4}$ are independent
differential transcendentals.

Consider the terms of these expressions ordered by some elimination ranking on
$\bar{\beta}$ with $\beta_{3}$ ranked highest. The leading term in this
ranking of each expression can be calculated using the definitions of previous
coefficients. The following table shows that these leading terms are non-zero:

$\begin{array}[]{c|l}A_{2,0}&\frac{-1}{\beta_{2}-\beta_{1}}\beta_{3}\\\
B_{1,1}&\frac{-2}{\beta_{2}-\beta_{3}}\beta_{3}^{\prime}\\\
A_{1,1}&\frac{-2}{(\beta_{2}-\beta_{1})B_{1,1}}\beta_{3}^{\prime\prime}\\\
B_{0,2}&\frac{-2}{(\beta_{2}-\beta_{1})A_{1,1}}\beta_{3}^{(3)}\end{array}$

We turn our attention to an elimination ranking with $\beta_{4}$ ranked
highest to prove that the remaining coefficients are nonzero. The following
table shows that the leading terms of these coefficients are also non-zero:

$\begin{array}[]{c|l}C_{2,2}&\frac{-3}{(\beta_{2}-\beta_{1})B_{0,2}}\beta_{4}^{\prime\prime}\\\
F_{1,1}&\frac{-4}{(\beta_{2}-\beta_{1})B_{0,2}C_{2,2}}\beta_{4}^{(3)}\\\
C_{1,3}&\frac{-7}{(\beta_{2}-\beta_{1})B_{0,2}F_{1,1}}\beta_{4}^{(4)}\\\
F_{0,2}&\frac{-7}{(\beta_{2}-\beta_{1})B_{0,2}C_{1,3}}\beta_{4}^{(5)}\end{array}$

We have shown that each substitution is well-defined, and therefore we have
constructed a definable bijection between system 2 and
$\mathbb{A}^{1}(\mathcal{U})$. Since $\mathbb{A}^{1}(\mathcal{U})$ has no
infinite rank subspaces, neither does our original system, completing the
proof of the proposition. ∎

We now finish the proof of Theorem 4.1. Since the differential tangent space
$T_{\bar{a}}^{\Delta}\left(X^{[4]}\right)$ satisfies the conditions of Lemma
4.2, it has no infinite rank subspaces over
$\mathbb{Q}\langle\bar{\alpha},\bar{a}\rangle$. Therefore, $X_{\alpha}$ is
strongly minimal by Theorem 3.8. ∎

###  4.2. Generic higher order equations

The technique used in the previous example can be applied to more general
classes of equations. In this section, we use analogous techniques to show
that generic equations with high enough degree have differential tangent
spaces cut out by linear equations with generic coefficients and that these
tangent spaces have no infinite rank subvariety.

Let

$f(x)=\alpha+\sum_{i=1}^{d}\alpha_{0,i}x^{i}+\sum_{j\in
M_{1}}\alpha_{1,j}m_{j}\left(x,x^{\prime}\right)+\cdots+\sum_{k\in
M_{h}}\alpha_{h,k}m_{k}\left(x,x^{\prime},\ldots,x^{(h)}\right)$

where $M_{n}$ indexes the set of all order $n$ monomials of degree at most $d$
and the entire collection of coefficients $\alpha,\alpha_{i,j}$ are
independent differential transcendentals over $\mathbb{Q}$. Let $V_{\alpha}$
be the zero set of $f(x)$ and let $m$ be the degree of nonmininality of $f$.
Following the notation of Section 3, we let $V^{[m]}$ be the following system
of equations in $x_{1},\ldots,x_{m},y$:

$\left\\{\begin{array}[]{lcl}\sum_{i=0}^{h}\sum_{j\in
M_{i}}\alpha_{i,j}m_{j}\left(x_{1},\ldots,x_{1}^{(i)}\right)&=&y\\\
\sum_{i=0}^{h}\sum_{j\in
M_{i}}\alpha_{i,j}m_{j}\left(x_{2},\ldots,x_{2}^{(i)}\right)&=&y\\\
&\vdots&\\\ \sum_{i=0}^{h}\sum_{j\in
M_{i}}\alpha_{i,j}m_{j}\left(x_{m},\ldots,x_{m}^{(i)}\right)&=&y\end{array}\right.$

Let $\bar{a}=(a_{1},\ldots,a_{m})$ be an indiscernible sequence in
$V_{\alpha}$ such that $a_{m}$ forks over $a_{1},\ldots,a_{m-1}$ and
$tp(a_{m}/\mathbb{Q}\langle\alpha,\alpha_{i,j},a_{1},\ldots,a_{m-1}\rangle_{i=0,\ldots,h,j\in
M_{i}}$ has rank between $1$ and $h-1$. That is, $a_{m}$ satisfies a
differential equation of order at least $1$ but no more than $h-1$ over
$\mathbb{Q}\langle\alpha,\alpha_{i,j},a_{1},\ldots,a_{m-1}\rangle_{i=0,\ldots,h,j\in
M_{i}}$. Crucially for this proof, we note that $a_{1},\ldots,a_{m}$ are
_algebraically_ independent over
$\mathbb{Q}\langle\alpha,\alpha_{i,j}\rangle$.

Let $T^{\Delta}_{(\bar{a},\alpha)}\left(V^{[m]}\right)$ denote the
differential tangent space of $V^{[m]}$ over $(\bar{a},\alpha)$. Then
$T^{\Delta}_{(\bar{a},\alpha)}\left(V^{[m]}\right)$ is given by

$\left\\{\begin{array}[]{lcl}\sum_{i=0}^{h}\beta_{i,1}z_{1}^{(i)}&=&y\\\
\sum_{i=0}^{h}\beta_{i,2}z_{2}^{(i)}&=&y\\\ &\vdots&\\\
\sum_{i=0}^{h}\beta_{i,m}z_{m}^{(i)}&=&y\end{array}\right.$

where $\beta_{i,j}=\frac{\partial f}{\partial x^{(i)}}(a_{j})$ as used in
previous sections.

######  Lemma 4.3.

Let $V$ be a differential variety cut out by a generic differential polynomial
of order $h$, degree $d$ and nonminimality degree $m$. If
$\bar{a}=(a_{1},\ldots,a_{m})$ is an indiscernible sequence such that $a_{m}$
satisfies an order $k$ equation over $a_{1},\ldots,a_{m}$ and the coefficients
of $f$ with $1<k<h$, then the variety
$T^{\Delta}_{(\bar{a},\alpha)}\left(V^{[m]}\right)$ has coefficients which are
independent differential transcendentals over $\mathbb{Q}$ whenever $d\geq
2m.$

###### Proof.

We proceed by induction on the order, beginning with
$\beta_{0,1},\beta_{0,2},\ldots,\beta_{0,m}$. Since $d\geq 2m,$ there are
$j_{1},\ldots,j_{m}$ such that $\alpha_{0,j_{1}},\ldots,\alpha_{0,j_{m}}$ are
independent differential transcendentals over
$\mathbb{Q}\left\langle\mathcal{A}_{0}a_{1}\cdots a_{m}\right\rangle$ where
$\mathcal{A}_{0}=\\{\alpha_{i,j}:0\leq i\leq h,j\in
M_{i}\\}\setminus\\{\alpha_{0,j_{1}},\ldots,\alpha_{0,j_{m}}\\}$. Note that

$\left(\begin{matrix}j_{1}a_{1}^{j_{1}-1}&j_{2}a_{1}^{j_{2}-1}&\ldots&j_{m}a_{1}^{j_{m}-1}\\\
j_{1}a_{2}^{j_{1}-1}&j_{2}a_{2}^{j_{2}-1}&\ldots&j_{m}a_{2}^{j_{m}-1}\\\
\vdots&&&\vdots\\\
j_{1}a_{m}^{j_{1}-1}&j_{2}a_{m}^{j_{2}-1}&\ldots&j_{m}a_{m}^{j_{m}-1}\end{matrix}\right)\left(\begin{matrix}\alpha_{0,j_{1}}\\\
\alpha_{0,j_{2}}\\\ \vdots\\\
\alpha_{0,j_{m}}\end{matrix}\right)=\left(\begin{matrix}\beta_{0,1}\\\
\beta_{0,2}\\\ \vdots\\\
\beta_{0,m}\end{matrix}\right)+\left(\begin{matrix}l_{1}\\\ l_{2}\\\ \vdots\\\
l_{m}\end{matrix}\right),$

where $l_{i}\in\mathbb{Q}\left\langle\mathcal{A}_{0}a_{1}\cdots
a_{m}\right\rangle$.

The above matrix is invertible, as the vanishing of its determinant imposes a
nontrivial algebraic relation among $a_{1},\ldots,a_{m}$, which are, by
assumption, algebraically independent. It follows that
$\beta_{0,1},\ldots,\beta_{0,m}$ are interdefinable with
$\alpha_{0,j_{1}},\ldots,\alpha_{0,j_{m}}$ over
$\mathbb{Q}\left\langle\mathcal{A}_{0}a_{1}\cdots a_{m}\right\rangle.$ Thus,
$\beta_{0,1},\ldots,\beta_{0,m}$ are independent and differentially
transcendental over $\mathbb{Q}\left\langle\mathcal{A}_{0}a_{1}\cdots
a_{m}\right\rangle.$ Note that the coefficients $\\{\beta_{i,j}:1\leq i\leq
h,1\leq j\leq m\\}$ are contained in the field
$\mathbb{Q}\left\langle\mathcal{A}_{0}a_{1}\cdots a_{m}\right\rangle,$ so
we’ve shown that $\beta_{0,1},\ldots,\beta_{0,m}$ are independent differential
transcendentals over $\mathbb{Q}\langle\beta_{i,j}:1\leq i\leq h,1\leq j\leq
m\rangle.$

Suppose we have already shown that $\beta_{n,1},\ldots,\beta_{n,m}$ are
independent differential transcendentals over
$\mathbb{Q}\langle\beta_{i,j}:n+1\leq i\leq h,1\leq j\leq m\rangle$ for some
$n<h$. Let $M^{*}_{n+1}$ index the collection of order $n+1$ monomials of
order no more than $d$ excluding the monomials of the form
$\\{x^{(n+1)}x^{r}:0\leq r\leq d-1\\}$. Assume the order $n+1$ terms in $f(x)$
are ordered so that

$\sum_{k\in
M_{n+1}}\alpha_{n+1,k}m_{k}(x,x^{\prime},\ldots,x^{(n+1)})=\sum_{k=0}^{d-1}\alpha_{n+1,k}x^{(n+1)}x^{k}+\sum_{j\in
M^{*}_{n+1}}\alpha_{n+1,j}m_{j}\left(x,x^{\prime},\ldots,x^{(n+1)}\right).$

Since $d\geq 2m,$ there are $k_{1},\ldots,k_{m}<d$ such that
$\alpha_{n+1,k_{1}},\ldots,\alpha_{n+1,k_{m}}$ are independent and
differentially transcendental over
$\mathbb{Q}\langle\mathcal{A}_{n+1}a_{1}\cdots a_{m}\rangle$ where
$\mathcal{A}_{n+1}=\\{\alpha_{i,j}:n+1\leq i\leq h,j\in
M_{i}\\}\setminus\\{\alpha_{n+1,k_{1}},\ldots,\alpha_{n+1,k_{m}}\\}$. Note
that

$\left(\begin{matrix}a_{1}^{k_{1}}&a_{1}^{k_{2}}&\ldots&a_{1}^{k_{m}}\\\
a_{2}^{k_{1}}&a_{2}^{k_{2}}&\ldots&a_{2}^{k_{m}}\\\ \vdots&&&\vdots\\\
a_{m}^{k_{1}}&a_{m}^{k_{2}}&\ldots&a_{m}^{k_{m}}\end{matrix}\right)\left(\begin{matrix}\alpha_{n+1,k_{1}}\\\
\alpha_{n+1,k_{2}}\\\ \vdots\\\
\alpha_{n+1,k_{m}}\end{matrix}\right)=\left(\begin{matrix}\beta_{n+1,1}\\\
\beta_{n+1,2}\\\ \vdots\\\
\beta_{n+1,m}\end{matrix}\right)+\left(\begin{matrix}r_{1}\\\ r_{2}\\\
\vdots\\\ r_{m}\end{matrix}\right),$

where $r_{i}\in\mathbb{Q}\langle\mathcal{A}_{n+1}a_{1}\cdots a_{m}\rangle$.
The above matrix is invertible, since we are assuming that
$a_{1},\ldots,a_{m}$ are algebraically independent. Thus,
$\alpha_{n+1,k_{1}},\ldots,\alpha_{n+1,k_{m}}$ are interdefinable with
$\beta_{n+1,1},\ldots,\beta_{n+1,m}$ over
$\mathbb{Q}\langle\mathcal{A}_{n+1}a_{1}\cdots a_{m}\rangle$. Since
$\\{\beta_{i,j}:n+1<i\leq h,1\leq j\leq m\\}$ are contained in the field
$\mathbb{Q}\langle\mathcal{A}_{n+1}a_{1}\cdots a_{m}\rangle$, it follows that
$\beta_{n+1,1},\ldots,\beta_{n+1,m}$ are independent and differentially
transcendental over $\\{\beta_{i,j}:n+1<i\leq h,1\leq j\leq m\\}$.

Putting together the above analysis, we have proved that the collection of
coefficients $\\{\beta_{i,j}:0\leq i\leq h,1\leq j\leq m\\}$ are independent
and differentially transcendental over $\mathbb{Q}\langle
a_{1},\ldots,a_{m}\rangle$ and thus over $\mathbb{Q}.$ ∎

Eliminating the variable $y$ from
$T^{\Delta}_{(\bar{a},\alpha)}\left(V^{[m]}\right)$ results in a system of
$m-1$ linear equations in $m$ variables as described in the following theorem.

######  Theorem 4.4.

Consider a system of $m-1$ linear differential equations of order $h$ in
variables $(z_{1},\ldots,z_{m})$ of the form:

(4)
$\left\\{\begin{array}[]{lcl}\sum_{i=0}^{h}\beta_{i,1}z_{1}^{(i)}&=&\sum_{i=0}^{h}\beta_{i,2}z_{2}^{(i)}\\\
\sum_{i=0}^{h}\beta_{i,1}z_{1}^{(i)}&=&\sum_{i=0}^{h}\beta_{i,3}z_{3}^{(i)}\\\
&\vdots&\\\
\sum_{i=0}^{h}\beta_{i,1}z_{1}^{(i)}&=&\sum_{i=0}^{h}\beta_{i,m}z_{m}^{(i)}.\end{array}\right.$

where the entire set of coefficients $\\{\beta_{i,j}:0\leq i\leq h,1\leq j\leq
m\\}$ are independent differential transcendentals. Then the solution set of
such a system has no infinite rank subspaces for $h>1$ and $m>1$.

As in Proposition 4.2, we show that such a system has no infinite rank
subspaces by constructing a definable bijection to
$\mathbb{A}^{1}(\mathcal{U})$. This is accomplished through two algorithms:
Algorithm A will apply a sequence of linear substitutions which will lower the
order of the top equation by one, and Algorithm B will apply Algorithm A
inductively to reduce the number of equations and variables by one.

Algorithm A:

We begin with a system of equations of the form

$\left\\{\begin{array}[]{lcl}\sum_{i=0}^{\ell}a_{i}u^{(i)}&=&\sum_{i=0}^{\ell}b_{i}z^{(i)}\\\
\sum_{i=0}^{h}c_{i}u^{(i)}+\sum_{i=0}^{h-1}e_{i}z^{(i)}&=&\sum_{i=0}^{h}\beta_{i}w^{(i)}\\\
\end{array}\right.$

with $1\leq\ell\leq h$. We lower the order of the top equation by applying a
trio of substitutions. The first substitution replaces the variable $u$ with
$\tilde{u}$ where $u=\tilde{u}+\left(\frac{b_{\ell}}{a_{\ell}}\right)z$ which
reduces the order of $z$ in the top equation by one. This results in the new
system:

$\left\\{\begin{array}[]{lcl}\sum_{i=0}^{\ell}a_{i}\tilde{u}^{(i)}&=&\sum_{i=0}^{\ell-1}\tilde{b}_{i}z^{(i)}\\\
\sum_{i=0}^{h}c_{i}\tilde{u}^{(i)}+\sum_{i=0}^{h}d_{i}z^{(i)}&=&\sum_{i=0}^{h}\beta_{i}w^{(i)}\\\
\end{array}\right.$

where

$\displaystyle d_{h}$
$\displaystyle:=c_{h}\left(\frac{b_{\ell}}{a_{\ell}}\right),$ $\displaystyle
d_{i}$
$\displaystyle:=e_{i}+\sum_{k=0}^{h-i}\binom{h}{h-i-k}c_{h-k}\left(\frac{b_{\ell}}{a_{\ell}}\right)^{(h-i-k)}$

for $0\leq i\leq h-1$, and

$\tilde{b}_{i}:=b_{i}-\sum_{k=0}^{\ell-i}\binom{\ell}{\ell-
i-k}a_{\ell-k}\left(\frac{b_{\ell}}{a_{\ell}}\right)^{(\ell-i-k)}$

for $0\leq i\leq\ell-1$.

The next substitution replaces $v$ with $\tilde{v}$, defined by
$v=\tilde{v}+\left(\frac{d_{h}}{\beta_{h}}\right)z$, reducing the order of $z$
in the lower equation by one. This results in the system of equations

$\left\\{\begin{array}[]{lcl}\sum_{i=0}^{\ell}a_{i}\tilde{u}^{(i)}&=&\sum_{i=0}^{\ell-1}\tilde{b}_{i}z^{(i)}\\\
\sum_{i=0}^{h}c_{i}\tilde{u}^{(i)}+\sum_{i=0}^{h-1}\tilde{e}_{i}z^{(i)}&=&\sum_{i=0}^{h}\beta_{i}\tilde{v}^{(i)}\\\
\end{array}\right.$

where

$\tilde{e}_{i}:=d_{i}-\sum_{k=0}^{h-i}\binom{h}{h-i-k}\beta_{h-k}\left(\frac{d_{h}}{\beta_{h}}\right)^{(h-i-k)}$

for $0\leq i\leq h-1$.

To complete the trio, we substitute $\tilde{z}$ for $z$ defined by
$z=\tilde{z}+\left(\frac{a_{\ell}}{\tilde{b}_{\ell-1}}\right)\tilde{u}^{\prime}$.
Now we have the system

$\left\\{\begin{array}[]{lcl}\sum_{i=0}^{\ell-1}\tilde{a}_{i}\tilde{u}^{(i)}&=&\sum_{i=0}^{\ell-1}\tilde{b}_{i}\tilde{z}^{(i)}\\\
\sum_{i=0}^{h}\tilde{c}_{i}\tilde{u}^{(i)}+\sum_{i=0}^{h-1}\tilde{e}_{i}\tilde{z}^{(i)}&=&\sum_{i=0}^{h}\beta_{i}\tilde{v}^{(i)}\\\
\end{array}\right.$

where

$\displaystyle\tilde{c}_{i}$
$\displaystyle:=c_{i}+\sum_{k=0}^{h-i}\binom{h-1}{h-i-k}\tilde{e}_{h-1-k}\left(\frac{a_{\ell}}{\tilde{b}_{\ell-1}}\right)^{(h-i-k)}$
$\displaystyle\tilde{c}_{0}$ $\displaystyle:=c_{0}$

for $1\leq i\leq h$ and

$\displaystyle\tilde{a}_{i}$
$\displaystyle:=a_{i}-\sum_{k=0}^{\ell-i}\binom{\ell-1}{\ell-
i-k}\tilde{b}_{\ell-1-k}\left(\frac{a_{\ell}}{\tilde{b}_{\ell-1}}\right)^{(\ell-
i-k)}$ $\displaystyle\tilde{a}_{0}$ $\displaystyle:=a_{0}$

for $1\leq i\leq\ell-1$.

We must first prove that each of these linear substitutions is well-defined,
i.e., that the coefficients $a_{\ell}$, $\beta_{h}$, and $\tilde{b}_{\ell-1}$
which appear in the denominators of substitutions are all non-zero.

######  Lemma 4.5.

The tuple $(b_{i},a_{i}:0\leq i\leq\ell)$ is interdefinable with
$(b_{\ell},a_{\ell},\tilde{b}_{i},\tilde{a}_{i}:0\leq i\leq\ell-1)$ as defined
in Algorithm A.

###### Proof.

It is clear from the definition of $\tilde{b}_{i}$ that $(b_{i}:0\leq
i\leq\ell-1)$ is interdefinable with $(\tilde{b}_{i}:0\leq i\leq\ell-1)$ over
$b_{\ell}$ and $\\{a_{i}:0\leq i\leq\ell\\}$. By adding the parameters
themselves, we have $(b_{i},a_{i}:0\leq i\leq\ell)$ is interdefinable with
$(b_{\ell},\tilde{b}_{i}:0\leq i\leq\ell-1)\cup(a_{i}:0\leq i\leq\ell)$. It is
also clear from the definition of $\tilde{a}_{i}$ that $(a_{i}:0\leq
i\leq\ell-1)$ is interdefinable with $(\tilde{a}_{i}:0\leq i\leq\ell-1)$ over
$a_{\ell}$ and $\\{\tilde{b}_{i}:0\leq i\leq\ell-1\\}$. Combining these two
facts, we conclude that the desired tuples are interdefinable. ∎

Assuming the initial coefficients $\\{a_{i},b_{i}:0\leq
i\leq\ell\\}\cup\\{\beta_{i}:0\leq i\leq h\\}$ constitute an independent set
of differential transcendentals, Lemma 4.5 shows that
$\\{\tilde{a}_{i},\tilde{b}_{i}:0\leq i\leq\ell-1\\}\cup\\{\beta_{i}:0\leq
i\leq h\\}\cup\\{a_{\ell},b_{\ell}\\}$ are also independent differential
transcendentals, and hence non-zero. It follows that the substitutions in
Algorithm A are well-defined as long as the initial coefficients have this
property.

######  Lemma 4.6.

Supposed that $(e_{i}:0\leq i\leq h-1)$ is differentially algebraic over
$\\{c_{i}:0\leq i\leq
h\\}\cup\\{a_{\ell},b_{\ell},\tilde{a}_{\ell-1},\tilde{b}_{\ell-1},\\}\cup\\{\beta_{i}:0\leq
i\leq h\\}$. Then $(c_{i}:0\leq i\leq h)$ is inter-differentially algebraic
with $(\tilde{c}_{i}:0\leq i\leq h)$ over $\\{a_{i},b_{i}:0\leq
i\leq\ell\\}\cup\\{\beta_{i}:0\leq i\leq\ell\\}$, and $(\tilde{e}_{i}:0\leq
i\leq h-1)$ is differentially algebraic over $\\{\tilde{c}_{i}:0\leq i\leq
h\\}\cup\\{a_{\ell},b_{\ell},\tilde{a}_{\ell-1},\tilde{b}_{\ell-1},\\}\cup\\{\beta_{i}:0\leq
i\leq h\\}$.

###### Proof.

From the definitions, it is clear that $(\tilde{c}_{i}:0\leq i\leq h)$ is
differentially algebraic over $(c_{i}:0\leq i\leq h)$. Now to prove the other
direction, we combine the definitions of $\tilde{c}_{i}$, $\tilde{e}_{i}$, and
$d_{i}$, we can write $\tilde{c}_{i}$ in terms of $(c_{i}:0\leq i\leq h)$ (as
well as other terms). Doing this for $\tilde{c}_{h}$ (and performing some
simplifications), we have

(5)
$\tilde{c}_{h}=\frac{b_{\ell}}{\tilde{b}_{\ell-1}}c_{h-1}-\frac{hb_{\ell}}{\tilde{b}_{\ell-1}}c_{h}^{\prime}+\left[1+\frac{a_{\ell}}{\tilde{b}_{\ell-1}}\left(h\left(\frac{b_{\ell}}{a_{\ell}}\right)^{\prime}-h\beta_{h}\left(\frac{b_{\ell}}{\beta_{h}a_{\ell}}\right)^{\prime}-\frac{\beta_{h-1}b_{\ell}}{\beta_{h}a_{\ell}}\right)\right]c_{h}+\frac{a_{\ell}}{\tilde{b}_{\ell-1}}e_{h-1}.$

Doing the same other $\tilde{c}$’s, we see that $c_{h}$ appears with order at
least one, but each $c_{i}$ is order zero and linear over $\mathbb{Q}\langle
a_{\ell},\tilde{a}_{\ell-1},b_{\ell},\tilde{b}_{\ell-1}\rangle$ for $i<h$.
This is represented in the following matrix equation:

(6) $\begin{bmatrix}M_{1}&M_{0}&0&\cdots&0&0&0&0\\\
M_{2}&M_{1}&M_{0}&\cdots&0&0&0&0\\\ M_{3}&M_{2}&M_{1}&\cdots&0&0&0&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots\\\
M_{h-3}&M_{h-4}&M_{h-5}&\cdots&M_{1}&M_{0}&0&0\\\
M_{h-2}&M_{h-3}&M_{h-4}&\cdots&M_{2}&M_{1}&M_{0}&0\\\
M_{h-1}&M_{h-2}&M_{h-3}&\cdots&M_{3}&M_{2}&M_{1}&M_{0}\\\
0&0&0&\cdots&0&0&0&1\end{bmatrix}\begin{bmatrix}c_{h-1}\\\ c_{h-2}\\\
c_{h-3}\\\ \vdots\\\ c_{3}\\\ c_{2}\\\ c_{1}\\\
c_{0}\end{bmatrix}=\begin{bmatrix}\tilde{c}_{h-1}\\\ \tilde{c}_{h-2}\\\
\tilde{c}_{h-3}\\\ \vdots\\\ \tilde{c}_{3}\\\ \tilde{c}_{2}\\\
\tilde{c}_{1}\\\ \tilde{c}_{0}\end{bmatrix}-\begin{bmatrix}L_{h-1}\\\
L_{h-2}\\\ L_{h-3}\\\ \vdots\\\ L_{3}\\\ L_{2}\\\ L_{1}\\\ 0\end{bmatrix}$

where
$L_{i}\in\mathbb{Q}\left\langle\\{c_{h},a_{\ell},b_{\ell},\tilde{a}_{\ell-1},\tilde{b}_{\ell-1}\\}\cup\\{\beta_{i}:0\leq
i\leq h\\}\cup\\{e_{i}:0\leq i\leq h-1\\}\right\rangle$. The matrix entries
can be computed as follows:

$\displaystyle M_{0}$ $\displaystyle=\frac{b_{\ell}}{\tilde{b}_{\ell-1}}$
$\displaystyle M_{1}$
$\displaystyle=h\left(\frac{b_{\ell}}{a_{\ell}}\right)^{\prime}\left(\frac{a_{\ell}}{\tilde{b}_{\ell-1}}\right)+(h-1)\left(\frac{b_{\ell}}{a_{\ell}}\right)\left(\frac{a_{\ell}}{\tilde{b}_{\ell-1}}\right)^{\prime}+1$
$\displaystyle M_{i}$
$\displaystyle=\sum_{k=0}^{i}\binom{h-1}{k}\binom{h}{i-k}\left(\frac{b_{\ell}}{a_{\ell}}\right)^{(i-k)}\left(\frac{a_{\ell}}{\tilde{b}_{\ell-1}}\right)^{(k)}$

for $i>1$. Using Gaussian elimination, we can make this matrix lower
triangular:

$M^{*}=\begin{bmatrix}M_{1,h-1}&0&0&\cdots&0&0&0&0\\\
M_{2,h-2}&M_{1,h-2}&0&\cdots&0&0&0&0\\\
M_{3,h-3}&M_{2,h-3}&M_{1,h-3}&\cdots&0&0&0&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots\\\
M_{h-3,3}&M_{h-4,3}&M_{h-5,3}&\cdots&M_{1,3}&0&0&0\\\
M_{h-2,2}&M_{h-3,2}&M_{h-4,2}&\cdots&M_{2,2}&M_{1,2}&0&0\\\
M_{h-1,1}&M_{h-2,1}&M_{h-3,1}&\cdots&M_{3,1}&M_{2,1}&M_{1,1}&0\\\
0&0&0&\cdots&0&0&0&1\end{bmatrix}$

where the new matrix entries can be computed recursively for $2\leq j\leq h-1$
and $1\leq i\leq h-j$

$M_{i,1}:=M_{i},\quad
M_{i,j}:=M_{i}-\left(\frac{M_{0}}{M_{1,j-1}}\right)M_{i+1,j-1}.$

######  Claim 4.7.

For all $1\leq j\leq h-1$ and $1\leq i\leq h-j$, $M_{i,j}\neq 0$, and
therefore, the determinant of $M^{*}$ is non-zero.

###### Proof.

We prove this by showing that $M_{i,j}$ is order $i+j-1$ in $a_{\ell}$ using
induction on $j$. It follows from Claim 4.5 that $M_{i,1}$ is order $i$ in
$a_{\ell}$ for each $i$. Now suppose the claim holds for $j$. Since
$M_{i,j+1}=M_{i}-\left(\frac{M_{0}}{M_{1,j}}\right)M_{i+1,j}$, we can see that
$M_{i,j+1}$ is order $i+j$ since $M_{i+1,j}$ is order $i+j$ by assumption, and
all other terms in the definition have strictly smaller order. Thus the
determinant is non-zero. ∎

Since $M^{*}$ is non-singular, it follows also that the original matrix in
Equation 6 is non-singular. Therefore, $(c_{i}:0\leq i\leq h-1)$ and
$(\tilde{c}_{i}:0\leq i\leq h-1)$ are interdefinable over
$\\{c_{h},a_{\ell},b_{\ell},\tilde{a}_{\ell-1},\tilde{b}_{\ell-1}\\}\cup\\{\beta_{i}:0\leq
i\leq h\\}\cup\\{e_{i}:0\leq i\leq h-1\\}$.

Turning our attention once more to Equation 5, we use this interdefinability
to substitute $c_{h-1}$ for an expression involving $c_{h}$ and
$(\tilde{c}_{i}:0\leq i\leq h)$. The resulting equation establishes a
differential relation between $c_{h}$ and $(\tilde{c}_{i}:0\leq i\leq h)$. It
follows that $(c_{i}:0\leq i\leq h)$ is differentially algebraic over
$\tilde{c}_{i}:0\leq i\leq h)$ along with
$\\{a_{\ell},b_{\ell},\tilde{a}_{\ell-1},\tilde{b}_{\ell-1}\\}\cup\\{\beta_{i}:0\leq
i\leq h\\}\cup\\{e_{i}:0\leq i\leq h-1\\}$. By assumption, $(e_{i}:0\leq i\leq
h-1)$ is identically 0 or differentially algebraic over $\\{c_{i}:0\leq i\leq
h\\}\cup\\{a_{\ell},b_{\ell},\tilde{a}_{\ell-1},\tilde{b}_{\ell-1},\\}\cup\\{\beta_{i}:0\leq
i\leq h\\}$. In either case, it follows that $(c_{i}:0\leq i\leq h)$ and
$\tilde{c}_{i}:0\leq i\leq h)$ are inter-differentially algebraic over
$\\{a_{\ell},b_{\ell},\tilde{a}_{\ell-1},\tilde{b}_{\ell-1}\\}\cup\\{\beta_{i}:0\leq
i\leq h\\}$.

So that this lemma can be applied inductively, we show that
$(\tilde{e}_{i}:0\leq i\leq h-1)$ is differentially algebraic over
$\\{\tilde{c}_{i}:0\leq i\leq
h\\}\cup\\{a_{\ell},b_{\ell},\tilde{a}_{\ell-1},\tilde{b}_{\ell-1},\\}\cup\\{\beta_{i}:0\leq
i\leq h\\}$. If $e_{i}=0$ for all $i$, this result follows from the definition
of $\tilde{e}_{i}$. Suppose otherwise, and we can see that $\tilde{e}_{i}$ is
defined by $(c_{i}:0\leq i\leq h)$, $(e_{i}:0\leq i\leq h-1)$, and
$\\{a_{\ell},b_{\ell},\tilde{b}_{\ell-1},\tilde{a}_{\ell-1}\\}\cup\\{\beta_{i}:0\leq
i\leq h\\}$. By assumption on $e_{i}$, we can see that $(\tilde{e}_{i}:0\leq
i\leq h-1)$ is differentially algebraic over $(c_{i}:0\leq i\leq h)$ and
$\\{a_{\ell},b_{\ell},\tilde{b}_{\ell-1},\tilde{a}_{\ell-1}\\}\cup\\{\beta_{i}:0\leq
i\leq h\\}$. We have already shown that, $(c_{i}:0\leq i\leq h)$ and
$(\tilde{c}_{i}:0\leq i\leq h)$ are inter-differentially algebraic, so
$(\tilde{e}_{i}:0\leq i\leq h-1)$ is differentially algebraic over
$(\tilde{c}_{i}:0\leq i\leq h)$, and
$\\{a_{\ell},b_{\ell},\tilde{b}_{\ell-1},\tilde{a}_{\ell-1}\\}\cup\\{\beta_{i}:0\leq
i\leq h\\}$. ∎

Algorithm B:

We begin with a system of equations of the form

(7)
$\left\\{\begin{array}[]{lcl}\sum_{i=0}^{h}a_{i}u^{(i)}&=&\sum_{i=0}^{h}b_{i}z^{(i)}\\\
\sum_{i=0}^{h}c_{i}u^{(i)}&=&\sum_{i=0}^{h}\beta_{i}w^{(i)}\\\
\end{array}\right.$

Algorithm B will eliminate the top equation in this system by first applying
Algorithm A $h$ times, reducing the order of the top equation to zero:

$\left\\{\begin{array}[]{lcl}\tilde{a}_{0}u&=&\tilde{b}_{0}z^{(i)}\\\
\sum_{i=0}^{h}\tilde{c}_{i}u^{(i)}+\sum_{i=0}^{h-1}\tilde{e}_{i}z^{(i)}&=&\sum_{i=0}^{h}\beta_{i}w^{(i)}\\\
\end{array}\right.$

Finally, $z=\frac{\tilde{a}_{0}}{\tilde{b}_{0}}u$, so we substitute this
expression for $z$ in the lower equation, eliminating $z$ from the system:

(8) $\sum_{i=0}^{h}\hat{c}_{i}u^{(i)}=\sum_{i=0}^{h}\beta_{i}w^{(i)}$

where

$\displaystyle\hat{c}_{h}$ $\displaystyle:=\tilde{c}_{h}$
$\displaystyle\hat{c}_{i}$
$\displaystyle:=\tilde{c}_{i}+\sum_{k=0}^{h-i}\binom{h-1}{h-i-k}\tilde{e}_{h-1-k}\left(\frac{\tilde{a}_{0}}{\tilde{b}_{0}}\right)^{(h-i-k)}+\cdots+\tilde{e}_{i}\left(\frac{\tilde{a}_{0}}{\tilde{b}_{0}}\right).$

######  Lemma 4.8.

For $a_{i}$, $b_{i}$, $c_{i}$, $\hat{c}_{i}$, and $\beta_{i}$ as defined in
Algorithm B, suppose $\\{a_{i},b_{i},\beta_{i}:0\leq i\leq h\\}$ is an
independent set of differential transendentals and suppose $(a_{i}:0\leq i\leq
h)$ and $(c_{i}:0\leq i\leq h)$ are inter-differentially algebraic over the
other coefficients $\\{b_{i},\beta_{i}:0\leq i\leq h\\}$. Then $(c_{i}:0\leq
i\leq h)$ and $(\hat{c}_{i}:0\leq i\leq h)$ are inter-differentially algebraic
over $\\{b_{i},\beta_{i}:0\leq i\leq h\\}$.

###### Proof.

By Lemma 4.6 and Lemma 4.5, we see that $(c_{i}:0\leq i\leq h)$ is inter-
differentially algebraic with $(\tilde{c}_{i}:0\leq i\leq h)$ over
$\\{a_{i},b_{i}:0\leq i\leq h\\}\cup\\{\beta_{i}:0\leq i\leq\ell\\}$. Since
$(a_{i}:0\leq i\leq h)$ is inter-differentially algebraic with $(c_{i}:0\leq
i\leq h)$ by assumption, it follows that $(c_{i}:0\leq i\leq h)$ is inter-
differentially algebraic with $(\tilde{c}_{i}:0\leq i\leq h)$ over
$\\{b_{i}:0\leq i\leq h\\}\cup\\{\beta_{i}:0\leq i\leq\ell\\}$. Consequently,
$(\tilde{c}_{i}:0\leq i\leq h)$ is independent over $\\{b_{i}:0\leq i\leq
h\\}\cup\\{\beta_{i}:0\leq i\leq\ell\\}$.

It follows from the definition that $(\tilde{c}_{i}:0\leq i\leq h)$ is
interdefinable with $(\hat{c}_{i}:0\leq i\leq h)$ over $\\{a_{i},b_{i}:0\leq
i\leq h,\\}\cup\\{\beta_{i}:0\leq i\leq h\\}\cup\\{\tilde{e}_{i}:0\leq i\leq
h-1\\}$. By Lemma 4.6, $(\tilde{e}_{i}:0\leq i\leq h-1)$ is differentially
algebraic over $\\{\tilde{c}_{i}:0\leq i\leq h\\}\cup\\{a_{i},b_{i}:0\leq
i\leq h\\}\cup\\{\beta_{i}:0\leq i\leq h\\}$, and we have already seen that
$(a_{i}:0\leq i\leq h)$ is inter-differentially algebraic with
$(\tilde{c}_{i}:0\leq i\leq h)$. Thus, $(\hat{c}_{i}:0\leq i\leq h)$ is
interdefinable with $(\tilde{c}_{i}:0\leq i\leq h)$, and hence with
$(c_{i}:0\leq i\leq h)$ over $\\{b_{i}:0\leq i\leq h\\}\cup\\{\beta_{i}:0\leq
i\leq\ell\\}$. ∎

######  Remark 4.9.

Due to the nature of the substitutions defining Algorithm A and Algorithm B,
only coefficients from the first equation influence the lower equation, and
not vice versa. For simplicity, Algorithm A and Algorithm B have been defined
for systems of two equations, although due to this lack of interaction, these
algorithms can be easily applied to systems of many equations. The only chance
that need be made is that an analogous form of the middle substitution in
Algorithm A must be applied to all lower equations simultaneously (lowering
the order of $z$ in all of these equations by 1).

Using Lemma 4.8, we can apply Algorithm B recursively to system 4 until all
but one variables have been eliminated. This results in a definable bijection
between system 4 and $\mathbb{A}^{1}(\mathcal{U})$, resulting in a proof of
Theorem 4.4.

Combining the results of Lemma 4.3, Theorem 4.4, and Theorem 3.8 results in a
proof of the main result, Theorem 1.3.

Finally, we will present an example of a system of equations which does not
have transcendental coefficients where the substitutions of algorithms A and B
are not well-defined.

######  Example 4.10.

Consider a generic fiber of the differential equation satisfied by the
$j$-function

$E_{\alpha}:=S(x)+R(x)(x^{\prime})^{2}=\alpha$

where $\alpha$ is a differential transcendental, $S(x)$ is the Schwarzian
derivative:

$S(x)=\left(\frac{x^{\prime\prime}}{x^{\prime}}\right)^{\prime}-\frac{1}{2}\left(\frac{x^{\prime\prime}}{x^{\prime}}\right)^{2}$

and

$R(x)=\frac{x^{2}-1978x+2654208}{2x^{2}(x-1728)^{2}}.$

Following the procedure laid our in Section 3, let $(a_{1},a_{2})$ be a Morley
sequence555If we were strictly applying the arguments from Section 3, we would
need a sequence of longer length. However, only two will be necessary to
demonstrate this example. for $E_{\alpha}$. We replace the transcendental
$\alpha$ with a variable $y$, compute the differential tangent space, and
eliminate $y$, resulting in the equation:

$A_{3}u^{(3)}+A_{2}u^{\prime\prime}+A_{1}u^{\prime}+A_{0}u=B_{3}v^{(3)}+B_{2}v^{\prime\prime}+B_{1}v^{\prime}+B_{0}v$

where

$\displaystyle A_{3}$ $\displaystyle:=\frac{1}{a_{1}^{\prime}},$
$\displaystyle B_{3}$ $\displaystyle:=\frac{1}{a_{2}^{\prime}},$
$\displaystyle A_{2}$
$\displaystyle:=-\frac{3a_{1}^{\prime\prime}}{(a_{1}^{\prime})^{2}},$
$\displaystyle B_{2}$
$\displaystyle:=-\frac{3a_{2}^{\prime\prime}}{(a_{2}^{\prime})^{2}},$
$\displaystyle A_{1}$
$\displaystyle:=-\frac{a_{1}^{(3)}}{(a_{1}^{\prime})^{2}}+\frac{3(a_{1}^{\prime\prime})^{2}}{(a_{1}^{\prime})^{3}}+2a_{1}^{\prime}R(a_{1}),$
$\displaystyle B_{1}$
$\displaystyle:=-\frac{a_{2}^{(3)}}{(a_{2}^{\prime})^{2}}+\frac{3(a_{2}^{\prime\prime})^{2}}{(a_{2}^{\prime})^{3}}+2a_{2}^{\prime}R(a_{2}),$
$\displaystyle A_{0}$ $\displaystyle:=(a_{1}^{\prime})^{2}R^{\prime}(a_{1}),$
$\displaystyle B_{0}$ $\displaystyle:=(a_{2}^{\prime})^{2}R^{\prime}(a_{2}).$

Unlike in Section 4, these coefficients are not independent differential
transcendentals. In fact, $A_{2}=3A_{3}^{\prime}$ and $B_{2}=3B_{3}^{\prime}$.
We now apply the sequence of substitutions found in Algorithm A. First
$(u,v)\mapsto(\tilde{u},v)$ where $\tilde{u}=u-\frac{B_{3}}{A_{3}}v$ resulting
in the equation

$A_{3}\tilde{u}^{(3)}+A_{2}\tilde{u}^{\prime\prime}+A_{1}\tilde{u}^{\prime}+A_{0}\tilde{u}=\tilde{B}_{2}v^{\prime\prime}+\tilde{B}_{1}v^{\prime}+\tilde{B}_{0}v$

where

$\displaystyle\tilde{B}_{2}$
$\displaystyle:=B_{2}-3A_{3}\left(\frac{B_{3}}{A_{3}}\right)^{\prime}-A_{2}\left(\frac{B_{3}}{A_{3}}\right),$
$\displaystyle\tilde{B}_{1}$
$\displaystyle:=B_{1}-3A_{3}\left(\frac{B_{3}}{A_{3}}\right)^{\prime\prime}-3A_{2}\left(\frac{B_{3}}{A_{3}}\right)^{\prime}-A_{1}\left(\frac{B_{3}}{A_{3}}\right),$
$\displaystyle\tilde{B}_{0}$
$\displaystyle:=B_{0}-A_{3}\left(\frac{B_{3}}{A_{3}}\right)^{(3)}-3A_{2}\left(\frac{B_{3}}{A_{3}}\right)^{\prime\prime}-3A_{1}\left(\frac{B_{3}}{A_{3}}\right)^{\prime}-A_{0}\left(\frac{B_{3}}{A_{3}}\right).$

The next substitution would replace $v$ with $\tilde{v}$ where
$v=\tilde{v}+\left(\frac{A_{3}}{\tilde{B}_{2}}\right)\tilde{u}^{\prime}$ which
would reduce the order of the $\tilde{u}$. However, this substitution is not
well-defined because $\tilde{B}_{2}=0$:

$\displaystyle\tilde{B}_{2}$
$\displaystyle=3B_{3}^{\prime}-3A_{3}\left(\frac{B_{3}}{A_{3}}\right)^{\prime}-3A_{3}^{\prime}\left(\frac{B_{3}}{A_{3}}\right)$
$\displaystyle=3B_{3}^{\prime}-3A_{3}\left(\frac{B_{3}^{\prime}A_{3}-A_{3}^{\prime}B_{3}}{A_{3}^{2}}\right)-\frac{3A_{3}^{\prime}B_{3}}{A_{3}}$
$\displaystyle=3B_{3}^{\prime}-3B_{3}^{\prime}+\frac{3A_{3}^{\prime}B_{3}}{A_{3}}-\frac{3A_{3}^{\prime}B_{3}}{A_{3}}$
$\displaystyle=0.$

While this specific substitution does not yield a definable bijection, one can
use another substitution
$\left(\tilde{v}=v-\frac{A_{3}}{\tilde{B}_{1}}\tilde{u}^{\prime\prime}\right)$,
and in this case, the fact that $\tilde{B}_{2}=0$ results in fewer
substitutions necessary to reach the desired result.

##  5\. Orthogonality to the constants

We’ve seen that bounds of [10] on the degree of nonminimality can be used in
conjunction with linearization techniques to establish the strong minimality
of general classes of differential equations. There are two main obstacles to
the wide application of these techniques for establishing strong minimality of
many classical nonlinear equations.

  1. (1)

The methods developed in the previous sections seem to require at least one
coefficient of the equation in question to be differentially transcendental.

  2. (2)

For a given equation, even of small order, the computations required to verify
strong minimality are quite involved.

In this section, we will show how the computational demands can be
significantly reduced if a weaker condition than strong minimality is the
goal: _$V$ is either strongly minimal or almost internal to the constants_. In
[10], it is shown that if $\operatorname{nmdeg}(p)>1,$ then $p$ is an isolated
type which is almost internal to a non-locally modular type. In differentially
closed fields, this means that $p$ is almost internal to the constant field
whenever $\operatorname{nmdeg}(p)>1.$ Thus, verifying the weaker condition
only requires an analysis of forking extensions over one new solution - the
computations need only involve two unknown variables in this case.

######  Example 5.1.

We will show the equation

$x^{\prime\prime}+x^{2}-\alpha=0,$

where $\alpha$ is a differential transcendental, is either strongly minimal or
internal to the constants. If the equation is not internal to the constants
and not strongly minimal, then by the results of [10] there is an
indiscernible sequence of length two $(x_{1},x_{2})$, which would satisfy the
system

$\left\\{\begin{array}[]{lcl}x_{1}^{\prime\prime}+x_{1}^{2}&=&\alpha\\\
x_{2}^{\prime\prime}+x_{2}^{2}&=&\alpha\end{array}\right.$

such that $x_{2}$ satisfies an order one equation over $x_{1}.$ Using the same
strategy as in the previous sections, we replace $\alpha$ with a variable $y$
in both equations, and then compute the differential tangent space:

$\left\\{\begin{array}[]{lcl}u^{\prime\prime}+2x_{1}u&=&y\\\
v^{\prime\prime}+2x_{2}v&=&y.\end{array}\right.$

Eliminating $y$, we are left with the single equation

$u^{\prime\prime}+2x_{1}u=v^{\prime\prime}+2x_{2}v.$

Consider the definable bijection given by the substitution $(u,v)\mapsto(w,v)$
where $u=w+v$. This transforms the above equation into

$w^{\prime\prime}+2x_{1}w=2(x_{2}-x_{1})v$

which can be solved for $v$ since $x_{1}\neq x_{2}$. Therefore the
differential tangent space has no infinite rank subvarieties, a contradiction,
so $x^{\prime\prime}+x^{2}-\alpha=0$ is either strongly minimal or almost
internal to the constants.

######  Question 5.2.

Varieties which are internal to the constants have certain stronger properties
that may, in general, allow one show via some additional argument the strong
minimality of specific equations. For instance, by [9], if $X$ is
nonorthogonal to the constants, then there are infinitely many co-order one
subvarieties of $X$. Thus, in this setting, showing strong minimality (after
an argument like that of the example above) is equivalent to ruling out co-
order one subvarieties. Are there interesting classes of equations in which
one can successfully employ this strategy?

After the completion of this work, the authors, together with Guy Casale and
Joel Nagloo, were able to give an affirmative answer to Question 5.2 in the
case of the equation satisfied by the $j$-function. This gives a new proof of
the main theorem of [11]: the differential equation satisfied by the
$j$-function,

$\left(\frac{y^{\prime\prime}}{y^{\prime}}\right)^{\prime}-\frac{1}{2}\left(\frac{y^{\prime\prime}}{y^{\prime}}\right)^{2}+(y^{\prime})^{2}\cdot\frac{y^{2}-1968y+2654208}{y^{2}(y-1728)^{2}}=0$

is strongly minimal. We also employ the strategy to establish the strong
minimality of several new equations. Nevertheless, we have left Question 5.2
as stated above, since we feel pursuing this approach is an important
direction for future research.

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