arXiv subjects

Marina Poulet

Publications and source records attributed to Marina Poulet.

5 recordsLinked to original sources

arXiv2025

Computing basis of solutions of any Mahler equation

Mahler equations arise in a wide range of contexts including the study of finite automata, regular sequences, algebraic series over Fp(z), and periods of Drinfeld modules. Introduced a century ago by K. Mahler to study the transcendence of certain complex numbers, they have recently been the subject of several works establishing a deep connection between such transcendence properties and the nature of their solutions. While numerous studies have investigated these solutions, existing algorithms can only compute them in specific rings: rational functions, power series, Puiseux series, or Hahn series. This paper solves the problem by providing an algorithm that computes a complete basis of solutions for any Mahler equation, along with a decomposition of each solution over the field of Puiseux series. Along the way, we describe an algorithm that computes a fundamental matrix of solutions for any Mahler system.

cs.SC↗

arXiv2025

Regular singular Mahler equations and Newton polygons

Though Mahler equations have been introduced nearly one century ago, the study of their solutions is still a fruitful topic for research. In particular, the Galois theory of Mahler equations has been the subject of many recent papers. Nevertheless, long is the way to a complete understanding of relations between solutions of Mahler equations. One step along this way is the study of singularities. Mahler equations with a regular singularity at 0 have rather "nice" solutions: they can be expressed with the help of Puiseux series and solutions of equations with constant coefficients. In a previous paper, the authors described an algorithm to determine whether an equation is regular singular at 0 or not. Exploiting information from the Frobenius method and Newton polygons, we improve this algorithm by significantly reducing its complexity, by providing some simple criterion for an equation to be regular singular at 0, and by extending its scope to equations with Puiseux coefficients.

cs.SC↗

arXiv2024

An algorithm to recognize regular singular Mahler systems

This paper is devoted to the study of the analytic properties of Mahler systems at 0. We give an effective characterisation of Mahler systems that are regular singular at 0, that is, systems which are equivalent to constant ones. Similar characterisations already exist for differential and (q-)difference systems but they do not apply in the Mahler case. This work fills in the gap by giving an algorithm which decides whether or not a Mahler system is regular singular at 0. In particular, it gives an effective characterisation of Mahler systems to which an analog of Schlesinger's density theorem applies.

cs.SC↗

arXiv2022

On the computation of the difference Galois groups of order three equations

In this paper we consider the problem of computing the difference Galois groups of order three equations for a large class of difference operators including the shift operator (Case S), the $q$-difference operator (Case Q), the Mahler operator (Case M) and the elliptic case (Case E). We show that the general problem can be reduced to several ancillary problems. We prove criteria to detect the irreducible and imprimitive Galois groups. Finally, we give a sufficient condition of differential transcendence of solutions of order three difference equations. We also compute the difference Galois group of an equation suggested by Wadim Zudilin.

math.NT↗

arXiv2022

A density theorem for the difference Galois groups of regular singular Mahler equations

The difference Galois theory of Mahler equations is an active research area. The present paper aims at developing the analytic aspects of this theory. We first attach a pair of connection matrices to any regular singular Mahler equation. We then show that these connection matrices can be used to produce a Zariski-dense subgroup of the difference Galois group of any regular singular Mahler equation.

math.NT↗