Algebraic independence and linear difference equations

Abstract

We consider pairs of automorphisms $(φ,σ)$ acting on fields of Laurent or Puiseux series: pairs of shift operators $(φ\colon x\mapsto x+h_1, σ\colon x\mapsto x+h_2)$, of $q$-difference operators $(φ\colon x\mapsto q_1x,\ σ\colon x\mapsto q_2x)$, and of Mahler operators $(φ\colon x\mapsto x^{p_1},\ σ\colon x\mapsto x^{p_2})$. Given a solution $f$ to a linear $φ$-equation and a solution $g$ to a linear $σ$-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 $σ$-Galois theory of linear $φ$-equations.

Theorems & Definitions (83)

• Theorem 1.1: Schäfke and Singer
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Conjecture 1.5: Loxton and van der Poorten
• Theorem 1.6
• Theorem 1.7
• Remark 2.1
• Definition 2.2: Definition 2.2 in OvWib
• Proposition 2.3