[Paper Review] Mahler's method in several variables I: The theory of regular singular systems

Boris Adamczewski, Colin Faverjon|arXiv (Cornell University)|Sep 13, 2018

semigroups and automata theory21 references4 citations

TL;DR

This paper develops a multivariable extension of Mahler's method for linear regular singular systems, proving that algebraic relations among values of Mahler functions at algebraic points can be lifted to algebraic relations among the functions themselves over $\overline{\mathbb{Q}}(\mathbf{z})$. It further establishes that values of Mahler functions associated with different systems are algebraically independent if their underlying matrix transformations are sufficiently distinct, generalizing earlier results in one variable and providing a foundation for applications in automata theory and base expansion problems.

ABSTRACT

This is the first part of a work devoted to the study of linear Mahler systems in several variables from the perspective of transcendence and algebraic independence. We prove two main results concerning systems that are regular singular at the origin. Our interest in Mahler's method comes from the possible applications of these results to old problems involving automata theory and which concern the expansion of both natural numbers and real numbers in integer bases. In particular, problems which involve finite automata and base change. Such applications are studied in Part II of this work.

Motivation & Objective

• To develop a comprehensive theory of regular singular linear Mahler systems in several variables from the perspective of transcendence and algebraic independence.
• To generalize the Nesterenko–Shidlovskii–Beukers–André lifting principle to the multivariable setting.
• To establish algebraic independence results for values of Mahler functions under different matrix transformations.
• To lay the theoretical groundwork for applications in automata theory and base expansion problems involving finite automata and base change.
• To address the long-standing conjecture of van der Poorten on independence of Mahler function values, providing the first general framework beyond sporadic examples.

Proposed method

• The authors introduce the concept of regular singular systems in several variables, defined as systems conjugated via a ramified meromorphic gauge transform to a constant invertible matrix system.
• They employ an axiomatic framework based on families of Mahler systems and multivariate exponential polynomials to analyze the structure of solutions and their algebraic relations.
• A new vanishing theorem is developed to control the behavior of polynomials evaluated at transformed points, using valuation theory and formalization of the field $\mathbb{L}$.
• The proof relies on iterated functional relations and the structure of coefficients $\tau_{j,\mathbf{k}}$ to show that vanishing at specific points implies vanishing of the entire function.
• Analytic gauge transformations are used to reduce the general case to a simpler one, with careful control of convergence and algebraic structure.
• The theory is applied to show that the ideal of algebraic relations between values of different Mahler functions is generated by pure relations within each system, implying independence across systems.

Experimental results

Research questions

• RQ1 Can algebraic relations among the values of multivariable Mahler functions at algebraic points be lifted to algebraic relations among the functions themselves over $\overline{\mathbb{Q}}(\mathbf{z})$?
• RQ2 Under what conditions do values of Mahler functions associated with different matrix transformations exhibit algebraic independence at algebraic points?
• RQ3 To what extent can the classical Nesterenko–Shidlovskii lifting principle be extended to the multivariable setting of regular singular Mahler systems?
• RQ4 How can the structure of regular singular systems be used to derive independence results for families of Mahler functions?
• RQ5 Can the theory be applied to solve problems in automata theory related to base change and finite automata expansions?

Key findings

• Any homogeneous algebraic relation over $\overline{\mathbb{Q}}$ between the values $f_1(\boldsymbol{\alpha}), \ldots, f_m(\boldsymbol{\alpha})$ of a regular singular Mahler system can be lifted to a similar algebraic relation over $\overline{\mathbb{Q}}(\mathbf{z})$ between the functions $f_1(\mathbf{z}), \ldots, f_m(\mathbf{z})$.
• The ideal of algebraic relations between values of Mahler functions associated with different systems is generated by the pure algebraic relations within each individual system, implying that cross-system relations are trivial.
• The independence result holds when the underlying matrix transformations are sufficiently distinct, generalizing earlier sporadic results by Nishioka and Masser.
• The theory applies to systems conjugated via ramified meromorphic gauge transforms to constant invertible matrices, a class referred to as regular singular.
• The results provide a foundational framework for studying transcendence and algebraic independence in multivariable Mahler systems, with implications for automata theory and base expansion problems.
• The paper establishes a multivariable analogue of the Shidlovskii-type theorem for regular singular systems, extending known results from differential and difference equations to the Mahler setting.