Arithmetic properties of the Taylor coefficients of differentially algebraic power series

Abstract

Let $f=\sum_{n=0}^\infty f_n x^n \in \overline{\mathbb Q}[[x]$ be a solution of an algebraic differential equation $Q(x,y(x), \ldots, y^{(k)}(x))=0$, where $Q$ is a multivariate polynomial with coefficients in $\overline{\mathbb Q}$. The sequence $(f_n)_{n\ge 0}$ satisfies a non-linear recurrence, whose expression involves a polynomial $M$ of degree $s$. When the equation is linear, $M$ is its indicial polynomial at the origin. We show that when $M$ is split over $\mathbb Q$, there exist two positive integers $δ$ and $ν$ such that the denominator of $f_n$ divides $δ^{n+1}(νn+ν)!^{2s}$ for all $n\ge 0\ $, generalizing a well-known property when the equation is linear. This proves in this case a strong form of a conjecture of Mahler that Pólya--Popken's upper bound $n^{\mathcal{O}(n\log(n))}$ for the denominator of $f_n$ is not optimal. This also enables us to make Sibuya and Sperber's bound $\vert f_n\vert_v\le e^{\mathcal{O}(n)}$, for all finite places $v$ of $\overline{\mathbb Q}$, explicit in this case. Our method is completely effective and rests upon a detailed $p$-adic analysis of the above mentioned non-linear recurrences. Finally, we present various examples of differentially algebraic functions for which the associated polynomial $M$ is split over $\mathbb Q$, among which are Weierstraß' elliptic $\wp$ function, solutions of Painlevé equations, and Lagrange's solution to Kepler's equation.