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Frobenius method for Mahler equations

Julien Roques

TL;DR

This work extends the classical Frobenius method to linear Mahler equations by embedding the problem in a Hahn-series—diff erence-differential framework. It defines a differential-difference algebraic setting, uses Newton polygons to classify slopes and exponents, and constructs gauge- and evaluation-objects that produce a basis of solutions at $z=0$ in the Hahn-series realm. The main contributions include a complete extension of Frobenius-type machinery to Mahler operators, an explicit factorization theory aligned with slopes, and a concrete example that demonstrates the method. The results provide a structured, computable approach to understanding the local behavior of Mahler solutions and open avenues for algorithmic analysis in automatic sequences and related areas.

Abstract

Using Hahn series, one can attach to any linear Mahler equation a basis of solutions at 0 reminiscent of the solutions of linear differential equations at a regular singularity. We show that such a basis of solutions can be produced by using a variant of Frobenius method.

Frobenius method for Mahler equations

TL;DR

This work extends the classical Frobenius method to linear Mahler equations by embedding the problem in a Hahn-series—diff erence-differential framework. It defines a differential-difference algebraic setting, uses Newton polygons to classify slopes and exponents, and constructs gauge- and evaluation-objects that produce a basis of solutions at in the Hahn-series realm. The main contributions include a complete extension of Frobenius-type machinery to Mahler operators, an explicit factorization theory aligned with slopes, and a concrete example that demonstrates the method. The results provide a structured, computable approach to understanding the local behavior of Mahler solutions and open avenues for algorithmic analysis in automatic sequences and related areas.

Abstract

Using Hahn series, one can attach to any linear Mahler equation a basis of solutions at 0 reminiscent of the solutions of linear differential equations at a regular singularity. We show that such a basis of solutions can be produced by using a variant of Frobenius method.

Paper Structure

This paper contains 29 sections, 19 theorems, 219 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Frobenius method for regular singular linear differential equations
  4. Hahn series and other useful rings
  5. The ring of Hahn series
  6. Structure of difference ring on the ring of Hahn series $\mathscr{H}_{R}$
  7. The differential-difference ring $\mathscr{H}_{\mathbb C(\lambda)}$
  8. The differential-difference ring $\mathscr{R}_{\lambda}$
  9. The difference ring $\mathscr{R}$
  10. Evaluating elements of $\mathscr{R}_{\lambda}$ at $\lambda = c \in \mathbb C^{\times}$
  11. Newton polygons, slopes and exponents
  12. Mahler equations and Mahler operators
  13. Newton polygons
  14. Slopes
  15. Characteristic equations and exponents
  16. ...and 14 more sections

Key Result

Theorem 4

We have attached to any exponent $\alpha$ and to any $m \in \{0,\ldots,m_{\alpha}-1\}$ a solution of $\mathcal{L}$. We obtain in this way a family of $n$$\mathbb C$-linearly independent solutions of $\mathcal{L}$.

Figures (1)

  • Figure 1: The Newton polygon of the Mahler operator \ref{['eq:un exemple bis']}. The $\bullet$ represent the points $(p^{i},\operatorname{val}_{z}(a_{i}))$ for $i \in \{0,1,2\}$.

Theorems & Definitions (45)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 4: Frobenius FrobMethodFrob
  • Remark 5
  • Remark 6
  • Remark 7
  • Example 8
  • Remark 9
  • Proposition 10
  • ...and 35 more