Ax-Schanuel for the first Painlevé equation

The two-variable first Painlevé equation $\partial_x^2 y = 6y^2 + x$ is the functional-parametric form of the classical first Painlevé equation $y'' = 6y^2 + t$: in a differentially closed field, its solutions are pairs $(x, y)$ in which $y$ is a Painlevé transcendent viewed as a function of $x$. Aslanyan (2021), developing a framework that converts Ax-Schanuel-type predimension inequalities into strong minimality and geometric triviality statements in $\mathrm{DCF}_0$, conjectured (Conjecture 3.15) that this equation satisfies such an inequality: for any $n$ non-constant solutions $(x_i, y_i)$ with $(x_i/x_j)^5 \neq 1$ for $i \neq j$, the transcendence degree over the constants satisfies $\operatorname{td}_C(\bar{x}, \bar{y}, \partial_{\bar{x}}\bar{y}) \ge 2n+1$, the only permitted exceptional relations being those generated by the equation's $\mu_5$-symmetry $(x, y) \mapsto (\zeta^2 x, \zeta y)$. A proof would be the first Ax-Schanuel-type theorem for a nonlinear differential equation outside the uniformization and linear settings. Conjecture 3.15 is open. The problem is to prove or refute Conjecture 3.15 as stated.

Background

A differentially closed field of characteristic zero, denoted $\mathrm{DCF}_0$, is a field $K$ equipped with a derivation $D$ (a map $D: K \to K$ preserving addition and satisfying the Leibniz rule $D(ab) = a\,Db + b\,Da$) in which every consistent system of differential polynomial equations over $K$ already has a solution; its field of constants is $C = \{a \in K : Da = 0\}$. For non-constant $x, y \in K$ one writes $\partial_x y := Dy/Dx$, the derivative of $y$ regarded as a function of $x$. When $K$ is a field of meromorphic functions of a variable $t$, a solution of a two-variable equation such as $\partial_x^2 y = 6y^2 + x$ is a pair of functions $(x(t), y(t))$ for which $y$ solves the corresponding ordinary differential equation as a function of $x$.

In model theory, a definable set is strongly minimal if every definable subset of it is finite or cofinite, so that its dimension theory is that of a pure algebraically closed field; it is geometrically trivial (equivalently, strictly disintegrated) if the algebraic closure of any tuple is the union of the algebraic closures of its entries, informally: distinct non-algebraic solutions admit no algebraic relations among themselves. The strongly minimal sets of $\mathrm{DCF}_0$ of differential order 1 (those defined by first-order differential equations) have been classified, but for order greater than 1 no classification is known. The guiding examples come from uniformization. The modular $j$-function, the $j$-invariant of elliptic curves viewed as a function on the complex upper half plane, satisfies a third-order algebraic differential equation $S(j) + R(j) \cdot (j')^2 = 0$, where $S(f) = f'''/f' - \frac{3}{2}(f''/f')^2$ is the Schwarzian derivative and $R$ is an explicit rational function. Freitag and Scanlon proved, building on modular Ax-Lindemann-Weierstrass-type transcendence results, that the solution set of this differential equation is strongly minimal and geometrically trivial (and not $\aleph_0$-categorical, meaning its countable models are not all isomorphic — the first such example). Pila and Tsimerman proved the corresponding Ax-Schanuel theorem with derivatives: if $(z_i, j_i)$, $i = 1, \dots, n$, satisfy the two-variable equation of $j$ and the $j_i$ are pairwise modularly independent (not linked by modular, i.e. Hecke, correspondences), then $\operatorname{td}_C(\bar{z}, \bar{j}, \partial_{\bar{z}}\bar{j}, \partial^2_{\bar{z}}\bar{j}) \ge 3n + 1$, where $\operatorname{td}_C$ is transcendence degree over the constants.

Aslanyan abstracted the shape of this inequality. For a two-variable differential equation $E(x, y)$ defined by a differential polynomial $f(x, y) = 0$ of order $m = \operatorname{ord}_Y f$ (the order of $f$ in the variable $y$), and a collection $\mathcal{P}$ of two-variable polynomials encoding the 'special' algebraic relations for $E$ (for $j$ these are the modular polynomials), $E$ has the $\mathcal{P}$-ASD property (Ax-Schanuel with Derivatives with respect to $\mathcal{P}$) if for all non-constant solutions $(x_i, y_i)$ whose $y_i$ are pairwise $\mathcal{P}$-independent (that is, $P(y_i, y_j) \neq 0$ for all $P \in \mathcal{P}$ and all $i \neq j$) one has $\operatorname{td}_C(x_i, y_i, \partial_{x_i}y_i, \dots, \partial^{m-1}_{x_i}y_i : 1 \le i \le n) \ge mn + 1$. Aslanyan proved that such a predimension inequality (even in a weaker Ax-Lindemann-Weierstrass form) forces the non-constant fibres of $E$ to be strongly minimal and geometrically trivial, with the induced structure on their Cartesian powers given by $\mathcal{P}$-special subvarieties; he recovered the Freitag-Scanlon theorem for $j$ this way, and, combining Ax-Schanuel with an Existential Closedness principle (the assertion that suitably generic systems of equations over the reduct acquire solutions inside it), classified the strongly minimal sets of the $j$-reduct (the expansion of the field $K$ by the solution set of the equation of $j$, with the derivation forgotten). He then asked which further equations carry such inequalities, observing that the two-variable analogues of the equations $Dy = r(y)$ with $r$ rational never do, and that the uniformizers of non-arithmetic genus-zero Fuchsian groups are natural candidates for the strongest form, in which $\mathcal{P}$ is trivial, i.e. $\mathcal{P} = \{X - Y\}$ — here a Fuchsian group is a discrete subgroup of $\mathrm{PSL}_2(\mathbb{R})$ acting on the complex upper half plane, its uniformizer is the single generator of the group's invariant meromorphic functions on that half plane (the analogue of $j$ for the quotient surface, available since the genus is zero), and 'non-arithmetic' means not commensurable with an arithmetic group.

The first Painlevé equation $D^2 y = 6y^2 + t$ is the most classical differential equation outside this uniformization world. Its solutions, the Painlevé first transcendents, have a well-understood one-variable model theory: the equation is strongly minimal (a classical result of differential algebra), and Nishioka proved algebraic independence of its solutions (strict disintegratedness). Its two-variable version is $\partial_x^2 y = 6y^2 + x$. The transformation $(x, y) \mapsto (\zeta^2 x, \zeta y)$, for $\zeta$ a fifth root of unity, carries solutions to solutions and forces the algebraic relation $y_i^5 = y_j^5$ between such paired solutions, so any Ax-Schanuel inequality for this equation must at least exclude these tuples. Aslanyan's Conjecture 3.15 asserts that this $\mu_5$-symmetry is the only obstruction.

Problem Statement

In a sufficiently saturated differentially closed field $\mathcal{K} = (K; +, \cdot, D) \models \mathrm{DCF}_0$ of characteristic zero with field of constants $C$, consider the two-variable first Painlevé equation

$$\partial_x^2 y = 6y^2 + x, \qquad \partial_x y := \frac{Dy}{Dx}.$$

Prove or refute the following conjecture (Aslanyan's Conjecture 3.15, 'Ax-Schanuel for the first Painlevé equation'). If $(x_i, y_i)$, $i = 1, \dots, n$, are solutions of this equation with all $x_i$ and $y_i$ non-constant (that is, outside $C$) and $(x_i/x_j)^5 \neq 1$ for all $i \neq j$, then

$$\operatorname{td}_C\bigl(\bar{x}, \bar{y}, \partial_{\bar{x}}\bar{y}\bigr) \ge 2n + 1,$$

where $\operatorname{td}_C$ is transcendence degree over the constants, $\bar{x} = (x_1, \dots, x_n)$, $\bar{y} = (y_1, \dots, y_n)$, and $\partial_{\bar{x}}\bar{y} = (\partial_{x_1}y_1, \dots, \partial_{x_n}y_n)$. The source's author remarks that one could in fact replace the $x$'s by the $y$'s in the condition $(x_i/x_j)^5 \neq 1$ ('as those are equivalent'), so that the conjecture states that the equation has the $\{X^5 - Y^5\}$-ASD property: an Ax-Schanuel predimension inequality whose only exceptional ('special') algebraic relations are those generated by the $\mu_5$-symmetry $(x, y) \mapsto (\zeta^2 x, \zeta y)$, $\zeta^5 = 1$, that is, the $y_i$ are pairwise independent with respect to $\mathcal{P} = \{X^5 - Y^5\}$ (no polynomial $P \in \mathcal{P}$ vanishes on any pair $(y_i, y_j)$ with $i \neq j$). This asserted equivalence is a remark of the source resting on the unproved premise that the $\mu_5$-relations are the only algebraic relations between solutions of the two-variable equation; the operative hypothesis of the conjecture is the condition $(x_i/x_j)^5 \neq 1$ on the $x_i$ displayed above. The conjecture is quantified over all $n$; the corresponding functional statement for complex meromorphic solutions transfers to the $\mathrm{DCF}_0$ statement, and the weaker Ax-Lindemann-Weierstrass special case $\operatorname{td}_C(\bar x) = 1$ does not imply the conjecture.

Aslanyan poses the same task, to find the correct Ax-Schanuel inequality, for the two-variable analogues of the other generic Painlevé equations, for which not even the exceptional family $\mathcal{P}$ is presently specified; those extensions, equally open, are context for this problem rather than part of its completion boundary.

References

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