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 : x → x + h1, σ : x → x + h2), of q-difference operators (φ : x → q1x, σ : x → q2x), and of Mahler operators (φ : x → x p 1 , σ : x → 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.

Mahler's method in several variables and finite automata

“…This conjecture was first proved by Bell and the first author in [2], while a different proof was given by Schäfke and Singer [57]. Very recently, the authors of [7] even proved a stronger result also conjectured by Loxton and van der Poorten [55]: a p-Mahler function f p (z) ∈ Q[[z]] and a q-Mahler function f q (z) ∈ Q[[z]] are algebraically independent over Q(z), unless one of them is rational. This result refines Cobham's theorem by expressing, in algebraic terms, the discrepancy between aperiodic automatic sets associated with multiplicatively independent input bases.…”

“…This result refines Cobham's theorem by expressing, in algebraic terms, the discrepancy between aperiodic automatic sets associated with multiplicatively independent input bases. The proof given in [7] is based on a suitable parametrized Galois theory associated with linear Mahler equations and follows the strategy initiated in [6].…”

“…The case r = 1 is a classical result (see, for instance, [52, Theorem 5.1.7]), while, as previously mentioned, the case r = 2 is much harder and was only recently proved in [7].…”

Skew-invariant curves and the algebraic independence of Mahler functions

“…There are three basic kinds of exceptional polynomials: linear polynomials, monomials, and (scalings of) Chebyshev polynomials. 2 A polynomial P is exceptional if it is linear or P is conjugate 3 to X N or to ±C N where N = deg(P ).…”

“…. , g n ∈ L each satisfy nontrivial τ -difference equations over K. That is, for some r ∈ N there are non-zero 2 For each positive integer N there is a unique monic polynomial C N which satisfies the functional equation C N (X + 1 X ) = X N + 1 X N . For us, a Chebyshev polynomial is a polynomial of the form C N for some N ≥ 2.…”