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 language | English |
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Pages (from-to) | 1899-1932 |
Number of pages | 34 |
Journal | Journal of the European Mathematical Society |
Volume | 26 |
Issue number | 5 |
DOIs | |
Publication status | Published - 2024 |
Keywords
- algebraic independence
- Linear difference equations
- Mahler functions
- q-difference
ASJC Scopus subject areas
- General Mathematics
- Applied Mathematics