Estimates for polynomials in logarithms of some rational numbers
type: Item
instance of: scholarly article
title: Estimates for polynomials in logarithms of some rational numbers
zbMATH Open document ID: 1204.11123
published in: Journal of Mathematical Sciences (New York)
publication date: 2008-10-28T00:00:00Z
review text: The Hermite-Padé approximations of the second type for an algebra generated by a generalized Nikishin system of Markov functions corresponding to an infinite branching graph are investigated. Arithmetical applications of this construction are given. Namely, lower estimates for polynomials with integer coefficients in logarithms of some rational numbers are obtained. These estimates partially refine some known results obtained earlier by the Siegel method. The author proves the following. Theorem 1. If \(r, p\), and \(q\) are natural numbers which satisfy the condition \(q > \kappa p^{N+1}\), where \(\kappa = \kappa_r(p) = 6^{rN}2^{r+N}\) and \(N = N_r(p) = (p+1)^r-1\), then, for any nonzero polynomial \(P\) of \(p\) variables which have integer rational coefficients and whose degree does not exceed \(r\), i.e., \(P(z_1, \ldots, z_p)\in\mathbb Z[z_1, \ldots, zp]\), \(P\not\equiv 0\), \(\deg P\leq r\), the following inequality holds: \[ | P(\theta_1,\ldots, \theta_p)| > \frac{c}{q^N}\frac 1{H^\mu}, \] where \(H = H(P)\) is the height of polynomial \(P\), exponent \(\mu\) is defined by the formula \[ \mu=\mu_r(p;q)=\frac{N\log q+\log\kappa}{\log q-\log(\kappa p^{N+1})}, \] and \(c\) is an effective positive constant depending only on the numbers \(r\) and \(p\).
reviewed by: Olaf Ninnemann
Mathematics Subject Classification ID: 11J86
zbMATH DE Number: 5358124
zbMATH Keywords: Hermite-Padé approximations of the second type
zbMATH Keywords: generalized Nikishin system of Markov functions
author: V. N. Sorokin
MaRDI profile type: Publication
full work available at URL: s10958-007-0390-0
OpenAlex ID: W1985458086
cites work: Hermite-Pade approximants for systems of Markov-type functions
cites work: Q3855260
cites work: On linear independence of values of generalized polylogarithms
DOI: 10.1007/S10958-007-0390-0
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description: scientific article; zbMATH DE number 5358124