An inequality for values of Koksma's functions of two algebraically dependent \(p\)-adic numbers
type: Item
instance of: scholarly article
title: An inequality for values of Koksma's functions of two algebraically dependent \(p\)-adic numbers
zbMATH Open document ID: 1389.11105
author: Tomislav Pejković
published in: Periodica Mathematica Hungarica
publication date: 2017-11-10T00:00:00Z
review text: For \(p\) be a prime number, let \(\mathbb{Q}_p\) denote the completion of the field of rational numbers \(\mathbb{Q}\) with respect to \(p\)-adic absolute value \(| \cdot |_p\). Denote by \(w_n\) Mahler's function on \(\mathbb{Q}_p\) and denote by \(w_n^*\) Koksma's function on \(\mathbb{Q}_p\). The author shows that if \(\xi\), \(\eta\in \mathbb{Q}_p\) are two algebraically dependent transcendental numbers and \(P(\xi, \eta)=0\), where \(P(x, y)\in \mathbb{Z}[x,y]\) is a non-zero irreducible polynomial over \(\mathbb{Q}\), then \(w_n^*(\xi)+1\leq M(w^*_{nN}(\eta)+1)\) and \(w_n^*(\eta)+1\leq N(w^*_{nM}(\xi)+1)\) for every positive integer \(n\). Moreover it is shown that these inequalities become equalities in a special case.
reviewed by: Yukio Ohkubo
Mathematics Subject Classification ID: 11J04
zbMATH DE Number: 6806128
zbMATH Keywords: Koksma's classification
zbMATH Keywords: p-adic numbers
zbMATH Keywords: Diophantine approximation
MaRDI profile type: Publication
full work available at URL: s10998-016-0139-y
OpenAlex ID: W2340243119
cites work: Q5719099
cites work: Q4829657
cites work: Q3413659
cites work: Q4138813
cites work: Über die Mahlersche Klasseneinteilung der transzendenten Zahlen und die Approximation komplexer Zahlen durch algebraische Zahlen
cites work: Zur Approximation der Exponentialfunktion und des Logarithmus. Teil II.
DOI: 10.1007/S10998-016-0139-Y
description: scientific article; zbMATH DE number 6806128