An \(ABC\) inequality for Mahler's measure
type: Item
instance of: scholarly article
title: An \(ABC\) inequality for Mahler's measure
zbMATH Open document ID: 1162.11033
author: Jeffrey D. Vaaler
published in: Monatshefte für Mathematik
publication date: 2008-08-19T00:00:00Z
review text: Let \(\overline{\mathbb Q}\) be an algebraic closure of \(\mathbb Q\), and for \(N \in \mathbb N\), let \(\phi_0, \phi_1, \dots, \phi_N \in \overline{\mathbb Q} [x]\) be polynomials with no common zeroes, satisfying \[ \phi_0 + \phi_1 + \dots + \phi_N =0, \] and spanning a \(\overline{\mathbb Q}\)-vector space of dimension \(N\) inside \(\overline{\mathbb Q} [x]\). Let \(\mathcal Z\) denote the set of all zeroes of \(\prod_{i=0}^N \phi_i\) in \(\overline{\mathbb Q}\). It is proved that \[ \log \mu(\phi_0) \leq \binom N 2 \sum_{\alpha \in \mathcal Z} \log H(\alpha) + \Delta_N (\deg \phi_1, \dots, \deg \phi_N), \] where \(\mu\) denotes the global Mahler measure of a polynomial, \(H\) the absolute Weil height of an algebraic number, and \(\Delta_N\) the maximal local logarithmic Mahler measure of the Wronskian of \(N\) normed polynomials of fixed degrees, where the roots vary on the unit circle. Only \(\Delta_2 (m,n) = \log(m+n)\) is explicitly known, for \(N \geq 3\) upper bounds for \(\Delta_N\) are given in Lemmas 4 and 5 of the paper. This result can be seen as an analogue of Mason's \(abc\)-theorem for Mahler's measure. The proof runs similar to that of the generalized \(abc\)-theorem for \(N+1\) summands, see e.g. \textit{W. D. Brownawell} and \textit{D. W. Masser} [Math. Proc. Camb. Philos. Soc. 100, 427--434 (1986; Zbl 0612.10010)]. A special challenge is the new analytical problem of estimating \(\Delta_N\).
reviewed by: Günter Lettl
Mathematics Subject Classification ID: 11G50
Mathematics Subject Classification ID: 11R09
zbMATH DE Number: 5313121
zbMATH Keywords: Mahler measure
zbMATH Keywords: Weil height
zbMATH Keywords: Wronskian determinant
zbMATH Keywords: subharmonic function
MaRDI profile type: Publication
full work available at URL: s00605-008-0536-2
OpenAlex ID: W2044767288
cites work: Vanishing sums in function fields
cites work: Old and new conjectured diophantine inequalities
cites work: Factorization of certain cyclotomic functions
cites work: On the zeros of the derivative of a polynomial
cites work: The hyperelliptic equation over function fields
cites work: Norm form equations. I
cites work: The Number of Irreducible Factors of a Polynomial. I
cites work: POLYNOMIAL IDENTITIES AND HAUPTMODULN
cites work: Diagonal equations over function fields
DOI: 10.1007/S00605-008-0536-2
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description: scientific article; zbMATH DE number 5313121