Algebraic independence and linear difference equations
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
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2020
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| _version_ | 1866912076838993920 |
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| author | Adamczewski, Boris Dreyfus, Thomas Hardouin, Charlotte Wibmer, Michael |
| author_facet | Adamczewski, Boris Dreyfus, Thomas Hardouin, Charlotte Wibmer, Michael |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_09266 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Algebraic independence and linear difference equations Adamczewski, Boris Dreyfus, Thomas Hardouin, Charlotte Wibmer, Michael Number Theory Group Theory 12H10, 39A06, 39A10, 39A13, 39A45, 11J81 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. |
| title | Algebraic independence and linear difference equations |
| topic | Number Theory Group Theory 12H10, 39A06, 39A10, 39A13, 39A45, 11J81 |
| url | https://arxiv.org/abs/2010.09266 |