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 $Q(x,y(x),\dots ,y^{(k)}(x))=0$, 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 $\delta$ and $\nu$ such that the denominator of $f_n$ divides ${\delta }^{n+1}(\nu n+\nu ){!}^{2s}$ 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 $|f_n{|}_v\le e^{\mathcal{O}(n)}$, 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.

AmS-Latex, 25 pages