Subspace theorems for homogeneous polynomial forms
type: Item
instance of: scholarly article
title: Subspace theorems for homogeneous polynomial forms
zbMATH Open document ID: 1131.11047
author: Pei-Chu Hu
author: Chungchun Yang
published in: Israel Journal of Mathematics
publication date: 2007-04-23T00:00:00Z
review text: The authors prove among other things the following extension of the Subspace Theorem. Let \(K\) be an algebraic number field and let \(M_K\) denote its set of places. Choose a set of normalized absolute values \(\| \cdot\| _{\rho}\) \((\rho\in M_K )\) such that the product formula \(\prod_{\rho\in M_K}\| x \| _{\rho}=1 \) for \(x\in K^*\) holds. For vectors \(\xi =(x_0,\ldots ,x_n)\in K^{n+1}\), \(\rho\in M_K\), put \(\| \xi \| _{\rho}:=\max_i\| x_i\| _{\rho}\). Let \(S\) a finite set of places of \(K\) containing all archimedean places, and for \(\rho\in S\), let \(f_{\rho,i}\) \((i=0,\ldots, n)\) be homogeneous polynomials of degree \(m\geq 1\) in \(K[X_0,\dots, X_n]\) such that (*) \(\{ \xi\in\overline{K}^{n+1}: f_{\rho ,i}(\xi )=0\;(i=0,\dots ,n)\}=\{ 0\} \) for \(\rho\in S\). Then for every \(\varepsilon >0\) there exist a finite number of hypersurfaces \(Y_1,\ldots ,Y_s\) of \(\overline{K}^{n+1}\) such that for all \(S\)-integral points \(\xi\) in \(K^{n+1}\setminus\bigcup_i Y_i\) we have \[ \prod_{\rho\in S}\;\prod_{i=0}^n \| f_{\rho ,i}(\xi )\| _{\rho} \geq \Bigl(\max_{\rho\in S}\| \xi\| _{\rho}\Bigr)^{-\varepsilon}. \] This is a variation on a result proved earlier by \textit{P. Corvaja} and \textit{U. Zannier} [Am. J. Math. 126, 1033--1055 (2004; Zbl 1125.11022)]. Further, the authors' method of proof is similar to that of Corvaja and Zannier. The authors give a reformulation of their result which shows the analogy with Shifman's conjecture from value distribution theory, recently established by \textit{M. Ru} [Am. J. Math. 126, 215--226 (2004; Zbl 1044.32009)]. Further, the authors state some problems, among which to generalize to higher degree polynomials an extension of the Subspace Theorem due to \textit{M. Ru} and \textit{P.-M. Wong} [Invent. Math. 106, 195--216 (1991; Zbl 0758.14007)]. Reviewer's remark: Independently of the authors, Ferretti and the reviewer obtained a generalization of the authors' result in which the solutions \(\xi\) are taken fr…
Mathematics Subject Classification ID: 11J68
zbMATH DE Number: 5145470
zbMATH Keywords: subspace theorem
zbMATH Keywords: homogeneous polynomials
zbMATH Keywords: \(S\)-integral points
zbMATH Keywords: hypersurfaces
zbMATH Keywords: analogy with Shifman's conjecture
reviewed by: Jan-Hendrik Evertse
MaRDI profile type: Publication
full work available at URL: s11856-006-0002-6
OpenAlex ID: W2157422459
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DOI: 10.1007/S11856-006-0002-6
description: scientific article; zbMATH DE number 5145470