On the differential transcendentality of the Morita \(p\)-adic gamma function
type: Item
instance of: scholarly article
title: On the differential transcendentality of the Morita \(p\)-adic gamma function
published in: European Journal of Mathematics
publication date: 2023-03-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/2109.11448
review text: Differential Galois theory is a theory which is analogous to classical Galois theory in that it studies Galois groups of algebraic differential equations. The so-called \textit{Picard-Vessiot} theory developed the concept of differential Galois groups for differential field extensions, where \textit{differential field} means that it is equipped with a derivation. And, just like in the usual algebraic case, one can define differential algebraic or transcendental elements, depending on whether they solve an algebraic differential equation or not. This clearly written article consists of the proof that the \(p\)-adic-valued version of Euler's Gamma function, aka the Morita \(p\)-adic Gamma function, extended to a holomorphic function on \(\mathbb{C}_p\), the complex \(p\)-adic number field, does not satisfy any algebraic differential equation whose coefficients are rational functions with coefficients in \(\mathbb{C}_p\). In the language of differential Galois theory, this is stated as the Morita Gamma function being differentially transcendental over the function field \(\mathbb{C}_p(X)\). This is a \(p\)-adic analogue to Hölder's Theorem that Euler's Gamma function is differentially transcendental over \(\mathbb{C}(X)\). The proof is obtained by explicit calculations in the ring of differential polynomials over the complex \(p\)-adic function field of rational functions \(\mathbb{C}_p(X)\), and relies on the fact that the field \(\mathbb{C}_p\) is algebraically closed.
reviewed by: Patrick Erik Bradley
Mathematics Subject Classification ID: 11S80
Mathematics Subject Classification ID: 12H05
zbMATH DE Number: 7659687
zbMATH Keywords: \(p\)-adic gamma function
zbMATH Keywords: analog of gamma function
zbMATH Keywords: differentially transcendental function
zbMATH Keywords: Hölder's theorem
author: Elżbieta Sowa
MaRDI profile type: Publication
OpenAlex ID: W3200361212
cites work: Picard-Vessiot Extensions of Real Differential Fields
cites work: Real Liouville Extensions
cites work: Real Liouvillian extensions of partial differential fields
cites work: Real and \(p\)-adic Picard-Vessiot fields
cites work: Q4750752
cites work: Topological Galois Theory
cites work: Algebraic Groups and Algebraic Dependence
cites work: Q4065637
cites work: Q4492826
cites work: Q4283149
DOI: 10.1007/S40879-023-00602-7
description: scientific article; zbMATH DE number 7659687