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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.
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https://hal.archives-ouvertes.fr/hal-03403976
Contributor : Charlotte Hardouin Connect in order to contact the contributor
Submitted on : Tuesday, October 26, 2021 - 1:58:26 PM
Last modification on : Wednesday, November 3, 2021 - 4:05:57 PM

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  • HAL Id : hal-03403976, version 1

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Boris Adamczewski, Thomas Dreyfus, Charlotte Hardouin, Michael Wibmer. ALGEBRAIC INDEPENDENCE AND LINEAR DIFFERENCE EQUATIONS. Journal of the European Mathematical Society, European Mathematical Society, In press. ⟨hal-03403976⟩

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