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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2508.21237 |
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| _version_ | 1866916924959490048 |
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| author | Di Vizio, Lucia Pellarin, Federico |
| author_facet | Di Vizio, Lucia Pellarin, Federico |
| contents | We prove an Ax-Lindemann-Weierstrass differential transcendence result for Euler's gamma function, namely that the functions $Γ(ν-ζ_1(ν)),\dots,Γ(ν-ζ_n(ν))$ are differentially independent over the field of rational functions in the variable $ν$, with coefficients in the field $k$ of $1$-periodic meromorphic functions over $\mathbb C$, as soon as $ζ_1,\dots,ζ_n$ determine a set of algebraic functions over $k$, stable by conjugation and pairwise distinct modulo $\mathbb Z$. \par To prove this result we use both the Galois theory of difference equations and the theory of a characteristic zero analog of the Carlitz module introduced by the second author in 2013. As an intermediate result we give an explicit description of the Picard-Vessiot rings and of the Galois groups associated to the operators in the image of the Carlitz module, using techniques inspired by the Carlitz-Hayes theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_21237 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Carlitz module and a differential Ax-Lindemann-Weierstrass theorem for the Euler gamma function Di Vizio, Lucia Pellarin, Federico Number Theory Complex Variables 39A10, 33B15, 11R58, 11G09 We prove an Ax-Lindemann-Weierstrass differential transcendence result for Euler's gamma function, namely that the functions $Γ(ν-ζ_1(ν)),\dots,Γ(ν-ζ_n(ν))$ are differentially independent over the field of rational functions in the variable $ν$, with coefficients in the field $k$ of $1$-periodic meromorphic functions over $\mathbb C$, as soon as $ζ_1,\dots,ζ_n$ determine a set of algebraic functions over $k$, stable by conjugation and pairwise distinct modulo $\mathbb Z$. \par To prove this result we use both the Galois theory of difference equations and the theory of a characteristic zero analog of the Carlitz module introduced by the second author in 2013. As an intermediate result we give an explicit description of the Picard-Vessiot rings and of the Galois groups associated to the operators in the image of the Carlitz module, using techniques inspired by the Carlitz-Hayes theory. |
| title | The Carlitz module and a differential Ax-Lindemann-Weierstrass theorem for the Euler gamma function |
| topic | Number Theory Complex Variables 39A10, 33B15, 11R58, 11G09 |
| url | https://arxiv.org/abs/2508.21237 |