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- Title
Algebraic independence and linear difference equations.
- Authors
Adamczewski, Boris; Dreyfus, Thomas; Hardouin, Charlotte; Wibmer, Michael
- Abstract
We consider pairs of automorphisms acting on fields of Laurent or Puiseux series: pairs of shift operators .W x 7 x C h1; W x 7 x C h2/, of q-difference operators .W x 7 q1x, W x 7 q2x/, and of Mahler operators .W x 7 xp1 ; W x xp2 /. Given a solution f to a linear -equation and a solution g to an algebraic -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 -Galois theory of linear -equations.
- Subjects
LINEAR differential equations; AUTOMORPHISMS; ALGEBRAIC independence; HYPERGEOMETRIC functions; GALOIS theory
- Publication
Journal of the European Mathematical Society (EMS Publishing), 2024, Vol 26, Issue 5, p1899
- ISSN
1435-9855
- Publication type
Article
- DOI
10.4171/JEMS/1316