Algebraic independence and linear difference equations

Boris Adamczewski, Thomas Dreyfus, Charlotte Hardouin, Michael Wibmer

Research output: Contribution to journalArticlepeer-review

Abstract

We consider pairs of automorphisms(Φσ) acting on fields of Laurent or Puiseux series: pairs of shift operators(Φ x →h1;σx +x h2) of q-difference operators(Φ x → q1x,σ x → q2x) and of Mahler operators (σ x → xp1 ; σ x → xp2 ) Given a solution f to a linear Φ-equation and a solution g to an algebraic Φ-equation, both transcendental, we show that f and g are algebraically independent over the field of rational functions, assuming that the corresponding parameters are sufficiently independent. As a consequence, we settle a conjecture about Mahler functions put forward by Loxton and van der Poorten in 1987. We also give an application to the algebraic independence of q-hypergeometric functions. Our approach provides a general strategy to study this kind of question and is based on a suitable Galois theory: the Φ-Galois theory of linear Φ-equations.

Original languageEnglish
Pages (from-to)1899-1932
Number of pages34
JournalJournal of the European Mathematical Society
Volume26
Issue number5
DOIs
Publication statusPublished - 2024

Keywords

  • algebraic independence
  • Linear difference equations
  • Mahler functions
  • q-difference

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics

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