Algebraic independence and linear difference equations

  • Boris Adamczewski

    Univ Lyon, Université Claude Bernard Lyon 1, CNRS, France
  • Thomas Dreyfus

    Université de Strasbourg, France
  • Charlotte Hardouin

    Université Paul Sabatier, Toulouse, France
  • Michael Wibmer

    Graz University of Technology, Austria
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Abstract

We consider pairs of automorphisms acting on fields of Laurent or Puiseux series: pairs of shift operators , of -difference operators , , and of Mahler operators . Given a solution to a linear -equation and a solution to an algebraic -equation, both transcendental, we show that and 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 -hypergeometric functions. Our approach provides a general strategy to study this kind of question and is based on a suitable Galois theory: the -Galois theory of linear -equations.

Cite this article

Boris Adamczewski, Thomas Dreyfus, Charlotte Hardouin, Michael Wibmer, Algebraic independence and linear difference equations. J. Eur. Math. Soc. (2023), published online first

DOI 10.4171/JEMS/1316