Arithmetic properties of the Taylor coefficients of differentially algebraic power series

TL;DR

The paper establishes sharp, effective divisibility bounds for the denominators of Taylor coefficients of differentially algebraic power series when the associated multivariate polynomial $M$ is split over $\mathbb Q$. By transforming nonlinear recurrences to a nonnegative-shift form and performing a detailed $p$-adic analysis, the authors prove that there exist computable integers $\delta$ and $\nu$ such that the denominator of $f_n$ divides $\delta^{n+1}(\nu n+\nu)^{2s}$, with $s$ the degree of the right-hand side polynomial. This yields explicit $v$-adic bounds, and, in the linear or split cases, strengthens Mahler’s and Pólya–Popken’s bounds, providing a concrete, constructive approach to understanding the arithmetic of DA power series. The work also demonstrates the applicability of these results through a broad set of examples, including Weierstraß’ $\wp$, Painlevé equations, elliptic and modular forms, and Kepler’s equation, illustrating the practical impact for number theory and special functions. Overall, the paper makes the bounds on denominators effectively computable and far more explicit in the split case, contributing to a deeper arithmetic understanding of DA series and their coefficients.

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.

Theorems & Definitions (8)

• Theorem 1
• Corollary 1
• Lemma 1
• proof
• Theorem 2
• proof
• Proposition 1
• proof