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Computing basis of solutions of any Mahler equation

Colin Faverjon, Marina Poulet

TL;DR

The paper presents a constructive algorithm to compute a complete basis of solutions to linear Mahler equations over ${\mathbf K}[[z]]$ and to decompose each solution into Puiseux, Hahn, and constant components. It introduces the notion of an admissible pair $(P,\Theta)$ to realize the Puiseux part, reduces the problem to finite-dimensional linear algebra via a carefully controlled window, and provides algorithms to compute the Puiseux matrix $P$, the Hahn matrix $H$, and the constant part $e_C$ of the fundamental solution matrix. By combining these components, the method produces a full fundamental matrix of solutions and a basis of solutions in the desired decomposed form, with explicit equations for the constituent Puiseux series. The approach applies to Mahler systems and is demonstrated with examples, including a Rudin–Shapiro case and Carlitz zeta-function, highlighting the method’s effectiveness and its connections to Galois theory and purity phenomena. This establishes a practical, constructive framework for solving Mahler equations in broad algebraic settings, enabling systematic computation of solution bases and their structural components.

Abstract

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.

Computing basis of solutions of any Mahler equation

TL;DR

The paper presents a constructive algorithm to compute a complete basis of solutions to linear Mahler equations over and to decompose each solution into Puiseux, Hahn, and constant components. It introduces the notion of an admissible pair to realize the Puiseux part, reduces the problem to finite-dimensional linear algebra via a carefully controlled window, and provides algorithms to compute the Puiseux matrix , the Hahn matrix , and the constant part of the fundamental solution matrix. By combining these components, the method produces a full fundamental matrix of solutions and a basis of solutions in the desired decomposed form, with explicit equations for the constituent Puiseux series. The approach applies to Mahler systems and is demonstrated with examples, including a Rudin–Shapiro case and Carlitz zeta-function, highlighting the method’s effectiveness and its connections to Galois theory and purity phenomena. This establishes a practical, constructive framework for solving Mahler equations in broad algebraic settings, enabling systematic computation of solution bases and their structural components.

Abstract

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.

Paper Structure

This paper contains 32 sections, 12 theorems, 107 equations, 4 algorithms.

Table of Contents

  1. Introduction
  2. Linear Mahler equations
  3. Linear Mahler systems
  4. Organization of the paper
  5. Applications and connections
  6. Properties of the Puiseux part of a fundamental matrix of solutions
  7. Equations, systems and modules
  8. Ramification order of an admissible pair
  9. Valuations of an admissible pair
  10. Proof of \ref{['item:support_Theta']} of Proposition \ref{['prop:incomplete_basis']}
  11. Proof of \ref{['item:valuation']} of Proposition \ref{['prop:incomplete_basis']}
  12. Laurent series expansion of $P$
  13. A recurrence formula for the blocks of columns
  14. The window $\{\nu,\ldots,\mu\}$
  15. Reduction to a finite dimensional problem
  16. ...and 17 more sections

Key Result

Theorem 1

Let ${\mathbf K}$ be an effective field of characteristic $0$. Algorithm algo:basis_solutions takes as input a Mahler equationWe assume that we are provided with a way to compute as many coefficients as needed in the power series expansion of $a_0,\ldots,a_m$. over ${\mathbf K}[[z]]$ of the form eq:

Theorems & Definitions (29)

  • Theorem 1
  • Remark
  • Theorem 2
  • Remark
  • Definition 3
  • Proposition 4
  • proof
  • Proposition 5
  • Remark 6
  • Proposition 7
  • ...and 19 more