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An algorithm to recognize regular singular Mahler systems

Colin Faverjon, Marina Poulet

TL;DR

The paper addresses recognizing regular singularity at 0 for p-Mahler systems by developing an effective, algorithmic criterion. It reduces the problem to a finite-dimensional linear-algebraic test on explicitly constructed spaces $ rak X_d$ derived from ramification and valuation data, and shows that regular singularity at 0 is equivalent to $ ext{dim } rak X_d=m$ for some $d o rak D_0$, yielding a computable constant gauge when true. The authors provide a concrete algorithm to compute a ramification index, the associated gauge, and Puiseux data, with a complexity analysis and practical implementations. They illustrate the method on size-1 systems, a second-order example, and Mahler systems arising from automata, and discuss open problems about the inverse-matrix case and operator changes. The work fills a gap by delivering an effective decision procedure for Mahler regular singularity at 0 and clarifies the structure of gauge transformations and solution spaces in this setting.

Abstract

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.

An algorithm to recognize regular singular Mahler systems

TL;DR

The paper addresses recognizing regular singularity at 0 for p-Mahler systems by developing an effective, algorithmic criterion. It reduces the problem to a finite-dimensional linear-algebraic test on explicitly constructed spaces derived from ramification and valuation data, and shows that regular singularity at 0 is equivalent to for some , yielding a computable constant gauge when true. The authors provide a concrete algorithm to compute a ramification index, the associated gauge, and Puiseux data, with a complexity analysis and practical implementations. They illustrate the method on size-1 systems, a second-order example, and Mahler systems arising from automata, and discuss open problems about the inverse-matrix case and operator changes. The work fills a gap by delivering an effective decision procedure for Mahler regular singularity at 0 and clarifies the structure of gauge transformations and solution spaces in this setting.

Abstract

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.

Paper Structure

This paper contains 19 sections, 19 theorems, 99 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Vector solutions of some Mahler equations
  3. The cyclic vector lemma
  4. Ramification index of vector solutions of Mahler systems
  5. Valuation of vector solutions of Mahler systems
  6. Coefficients of vector solutions of Mahler systems
  7. A characterisation of regular singular Mahler systems at 0
  8. A concrete algorithm for Theorem \ref{['thm:algo']}
  9. Description of an algorithm computing a ramification index
  10. Description of the algorithm of Theorem \ref{['thm:Espace']}
  11. On the shape of Md and Nd
  12. Proof of Proposition \ref{['prop:complexity']}
  13. Examples
  14. Systems of size 1
  15. An equation of order 2
  16. ...and 4 more sections

Key Result

Theorem 1.3

Let $A \in {\rm GL}_m\left(\overline{\mathbb Q}(z)\right)$ and $p \geq 2$. There exists an algorithm which determines whether or not the Mahler system eq:Mahler_at_0 is regular singular at $0$. This is done by computing the dimension of an explicit $\overline{\mathbb Q}$-vector space. If the system

Theorems & Definitions (44)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 2.1: cyclic vector lemma
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • ...and 34 more