A purity theorem for Mahler equations

TL;DR

This paper establishes a purity theorem for Mahler functions, showing how solutions' arithmetic growth properties propagate within minimal Mahler equations, extending concepts from G- and E-functions.

Contribution

It introduces a new purity theorem for Mahler functions, linking solution properties to a natural filtration based on coefficient growth, and studies the structure of solutions via generalized Mahler series.

Findings

Solutions form a basis of generalized Mahler series.

Membership in the largest filtration pieces propagates among solutions.

The theorem does not hold for the smallest filtration pieces.

Abstract

The principal aim of this paper is to establish a purity theorem for Mahler functions that is reminiscent of famous purity theorems for G-functions by D. and G. Chudnovsky and for E-functions (and, more generally, for holonomic arithmetic Gevrey series) by Y. Andr{\'e}. Our approach is based on a preliminary study of independent interest of the nature of the solutions of Mahler equations. Roughly speaking, we prove a reduction result for Mahler systems, implying that any Mahler equation admits a complete basis of solutions formed of what we call generalized Mahler series. These are sums involving Puiseux series, Hahn series of a very special type and solutions of inhomogeneous equations of order 1 with constant coefficients. In the light of B. Adamczewski, J. P. Bell and D. Smertnig's recent height gap theorem, we introduce a natural filtration on the set of generalized Mahler series according to the arithmetic growth of the coefficients of the Puiseux series involved in their decomposition. This filtration has five pieces. Our purity theorem states that the membership of a generalized Mahler series to one of the three largest pieces of this filtration propagates to any other generalized Mahler series solution of its minimal Mahler equation. We also show that this statement does not extend to the smallest two pieces.