Arithmetic properties of the Taylor coefficients of differentially algebraic power series

TL;DR

This paper investigates the arithmetic properties of Taylor coefficients of differentially algebraic power series, establishing bounds on denominators and sizes, especially when associated polynomials are split over rationals, with applications to special functions.

Contribution

It proves new bounds on denominators of Taylor coefficients for a class of differentially algebraic functions when certain polynomials are split over rationals, extending known linear cases.

Findings

Denominator of coefficients divides a specific factorial-based expression.

Bounds on the size of coefficients at finite places are made explicit.

Includes examples like elliptic functions and solutions to Painlevé equations.

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 $\delta$ 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 P\'olya--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 Weierstra\ss' elliptic $\wp$ function, solutions of Painlev\'e equations, and Lagrange's solution to Kepler's equation.