Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hahn series and Mahler equations: Algorithmic aspects

C. Faverjon, Julien Roques

TL;DR

This work develops an algorithm to compute Hahn series solutions of linear Mahler equations by embedding the problem into a finite-dimensional linear-algebra framework. Central to the method is a computable, well-ordered receptacle $\mathcal{V}$ for all possible supports of Hahn series solutions, built from Newton polygon data and the maps $\Psi$, $\psi$, and $\pi$, together with a finite truncation set $\mathcal{R} \subset \mathcal{V}$ preserving the solution space via an isomorphism when a technical condition $\star_{\mathcal{V}}$ is met. The authors provide detailed algorithms to (i) compute $\mathcal{V}$ and $\mathcal{R}$, (ii) bound gaps $\epsilon(v)$ and $\tau$, and (iii) recover a basis for the space of solutions through explicit linear systems derived from the Mahler operator. The approach is illustrated by the Rudin–Shapiro Mahler equation, where the method yields explicit Hahn-series coefficients and confirms the expected dimension of the solution space. Overall, the paper delivers a constructive answer to whether Hahn-series solutions can be algorithmically calculated, connecting Newton polygon theory with computable support-truncation to enable precise, finite-linear-algebra computations in the Hahn-series setting.

Abstract

Many articles have recently been devoted to Mahler equations, partly because of their links with other branches of mathematics such as automata theory. Hahn series (a generalization of the Puiseux series allowing arbitrary exponents of the indeterminate as long as the set that supports them is well-ordered) play a central role in the theory of Mahler equations. In this paper, we address the following fundamental question: is there an algorithm to calculate the Hahn series solutions of a given linear Mahler equation? What makes this question interesting is the fact that the Hahn series appearing in this context can have complicated supports with infinitely many accumulation points. Our (positive) answer to the above question involves among other things the construction of a computable well-ordered receptacle for the supports of the potential Hahn series solutions.

Hahn series and Mahler equations: Algorithmic aspects

TL;DR

This work develops an algorithm to compute Hahn series solutions of linear Mahler equations by embedding the problem into a finite-dimensional linear-algebra framework. Central to the method is a computable, well-ordered receptacle for all possible supports of Hahn series solutions, built from Newton polygon data and the maps , , and , together with a finite truncation set preserving the solution space via an isomorphism when a technical condition is met. The authors provide detailed algorithms to (i) compute and , (ii) bound gaps and , and (iii) recover a basis for the space of solutions through explicit linear systems derived from the Mahler operator. The approach is illustrated by the Rudin–Shapiro Mahler equation, where the method yields explicit Hahn-series coefficients and confirms the expected dimension of the solution space. Overall, the paper delivers a constructive answer to whether Hahn-series solutions can be algorithmically calculated, connecting Newton polygon theory with computable support-truncation to enable precise, finite-linear-algebra computations in the Hahn-series setting.

Abstract

Many articles have recently been devoted to Mahler equations, partly because of their links with other branches of mathematics such as automata theory. Hahn series (a generalization of the Puiseux series allowing arbitrary exponents of the indeterminate as long as the set that supports them is well-ordered) play a central role in the theory of Mahler equations. In this paper, we address the following fundamental question: is there an algorithm to calculate the Hahn series solutions of a given linear Mahler equation? What makes this question interesting is the fact that the Hahn series appearing in this context can have complicated supports with infinitely many accumulation points. Our (positive) answer to the above question involves among other things the construction of a computable well-ordered receptacle for the supports of the potential Hahn series solutions.

Paper Structure

This paper contains 42 sections, 39 theorems, 250 equations, 2 figures, 6 algorithms.

Table of Contents

  1. Introduction
  2. Outline of our answer to Question \ref{['main question: un algo pour sol hahn?']}
  3. Organization of the paper
  4. The ring of Hahn series
  5. Newton polygons
  6. Newton polygon
  7. Slopes
  8. Newton polygon and slopes in the homogeneous case
  9. The maps $\Psi$, $\psi$ and $\pi$
  10. Supports, valuations and the maps $\Psi$, $\psi$ and $\pi$
  11. A map of fundamental importance
  12. A receptacle $\mathcal{V}$ for the support of the solutions
  13. Proof of the fact that $(\mathcal{V}_{i})_{i\geq 0}$ is nondecreasing in Theorem \ref{['theo: un support bis']}
  14. Basic properties of $\mathcal{V}$
  15. Proof of \ref{['supp sol dans S bis']} of Theorem \ref{['theo: un support bis']}
  16. ...and 27 more sections

Key Result

Lemma 3

The following properties relative to $\mu \in \mathbb Q$ are equivalent: Moreover, if $\mu$ belongs to $\mathcal{S}(L,a_{-\infty} )$, then the equality eq:caract slopes is satisfied if and only if $(\ell^{i_{1}},\operatorname{val} a_{i_1})$ and $(\ell^{i_{2}},\operatorname{val} a_{i_2})$ belong to the edge of slope $\mu$ of $\mathcal{N}(L,a_{-\infty} )$.

Figures (2)

  • Figure 1:
  • Figure 2: This figure is relative to the operator $L$ given by \ref{['L pour example']}. We have ${\mathcal{P}}(L)=\{p_{0},p_{1},p_{2},p_{3}\}$. The Newton polygon $\mathcal{N}(L)$ is the shaded area. Its vertices are $p_{0},p_{1}$ and $p_{2}$ and we have $\mathcal{S}(L)=\{0,1/2\}$. In each subfigure, we consider a specific $v \in \mathbb Q$. The point $p_{k}'$ is the projection of $p_{k}$ along a line of slope $-v$ onto the $y$-axis; so, the dotted segments have slope $-v$ and $\Psi(v)=\{p_0',p_1',p_2',p_3'\}$. The dashed segment with left extremity $p_{k}'$ is the segment with lowest slope among those linking $p_{k}'$ to an element of ${\mathcal{P}}(L)$. The slope of this segment is thus the opposite of $\pi(q_{k})$ where $q_{k}$ is the ordinate of $p_{k}'$ and, hence, $\pi(\Psi(v))$ is the set of the opposite of the slopes of the four dashed segments.

Theorems & Definitions (95)

  • Remark 2
  • Lemma 3
  • proof
  • Remark 4
  • Example 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 85 more