On algebraic relations in non-archimedean fields
type: Item
instance of: scholarly article
title: On algebraic relations in non-archimedean fields
zbMATH Open document ID: 0797.11062
DOI: 10.1007/BF01077066
publication date: 1994-04-12T00:00:00Z
review text: The author proves the following effective theorem: suppose that the series \(f_ 1(z), \dots, f_ m(z)\) belong to the class \(F(\mathbb K, c_ 1, c_ 2, c_ 3, q_ 0)\) and form a solution of a Siegel normal system. Let \(\xi \in \mathbb K\), \(\xi T(\xi) \neq 0\), \(\delta>0\). Then there exists an effective constant \(H_ 0\) such that for any polynomial \(Q(y_ 1, \dots, y_ n) \not \equiv 0\) of degree \(d\) and the height \(\leq H\), there exists for \(H \geq H_ 0\) a prime number \(p\) such that \[ p \leq (H \delta) {m + \delta \choose m} {m \log H \over \log \log H} \] and a place \(v \mid p\) such that in \(\mathbb K_ v\) the relation \(Q(f_ 1(\xi), \dots, f_ m (\xi)) \neq 0\) holds. The author also gives an application to hypergeometric series. The proof uses a modification of Siegel-Shidlovskii and the work by \textit{F. Beukers}, \textit{W. D. Brownawell} and \textit{G. Heckman} [Ann. Math. (2) 127, 279--308 (1988; Zbl 0652.10027)].
reviewed by: Xu Guangshan
Mathematics Subject Classification ID: 11J91
Mathematics Subject Classification ID: 11J61
zbMATH DE Number: 525643
zbMATH Keywords: \(F\)-function
zbMATH Keywords: \(p\)-adic field
zbMATH Keywords: algebraic relations
zbMATH Keywords: Siegel-Shidlovskii method
zbMATH Keywords: Siegel normal system
zbMATH Keywords: hypergeometric series
author: V. G. Chirskii
MaRDI profile type: Publication
cites work: Remarks on the perturbation theory for problems of Mathieu type
cites work: Three coupled oscillators: Mode-locking, global bifurcations and toroidal chaos
cites work: Resonance regions for Mathieu type dynamical systems on a torus
cites work: Resonance regions for families of torus maps
full work available at URL: https://doi.org/10.1007/bf01077066
OpenAlex ID: W1976765258
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 525643