Integral points of projective spaces omitting hyperplanes over function fields of positive characteristic
type: Item
instance of: scholarly article
title: Integral points of projective spaces omitting hyperplanes over function fields of positive characteristic
zbMATH Open document ID: 1022.11032
author: Julie Tzu-Yueh Wang
published in: Journal of Number Theory
publication date: 2003-10-26T00:00:00Z
review text: Let \(K\) be the function field of an irreducible smooth projective curve \(C\) of genus \(g\) defined over an algebraically closed field \(k\) of characteristic \(p>0\). The author proves some results concerning \(S\)-integral points in \(\mathbb{P}^n(K)\) minus \(2n+2\) hyperplanes. The precise statement is given below. The height of \((f_0,\ldots,f_n)\in \mathbb{P}^n(K)\) is given by \[ h(f_0,\ldots ,f_n)=-\sum_{P\in C(k)} \min ( v_P(f_0),\ldots ,v_P(f_n)), \] where \(v_P\) is the normalized valuation at \(P\). The height of \(a\in K\) is defined by \(h(a)=-\sum_{P\in C(k)}\min (0,v_P(a))\). Define \(2n+2\) linear forms \(L_0,\ldots ,L_{2n+1}\) in \(n+1\) variables with coefficients in \(K\) as follows: \(L_i =X_i\) for \(i=0,\ldots ,n\), and \(L_{n+1+i}=\sum_{i=0}^n a_{ij}X_j\) with \(a_{ij}\in K\). Suppose that \(L_0,\ldots ,L_n\) are in general position. Further suppose there is a positive integer \(m\) such that (*) the numbers \(\prod_{i=0}^n a_{i,\sigma (i)}\), where \(\sigma\) runs through the permutations of \(0,\ldots, n\), are linearly independent over \(K^{p^m}\). Let \(\widetilde{S}\) be the set of \(P\in C(k)\) such that \(v_P(\prod_{0\leq i,j\leq n} a_{ij})\not= 0\). Then for all \(S\)-integral points \((f_0,\ldots ,f_n)\in \mathbb{P}^n(K)\backslash \bigcup_{i=0}^{2n+1}\{ L_i=0\}\) one has \[ h(f_0,\ldots ,f_n)\leq {n(n+1)\over 2}p^{m-1}\max ( 0, 2g-2+|S\cup \widetilde {S}|) +\sum_{0\leq i,j\leq n} h(a_{ij}) . \] As a consequence the author shows that the number of \(S\)-integral points (up to constant factors in each coordinate) in \(\mathbb{P}^n(K)\backslash \cup_{i=0}^{2n+1}\{ L_i=0\}\) is at most \[ \Big( n(n+1)p^{m-1}\max (0,2g-2+|S\cup \widetilde{S}|)+1\Big)^{(n+1)|S|} . \] The author gives an example showing that the technical condition (*) cannot be removed completely. The author's main tool is her \(n\)-dimensional generalization [J. Number Theory 58, 139-157 (1996; Zbl 0866.11066)] of \textit{R. C. Mason}'s abc-theorem over function fields…
Mathematics Subject Classification ID: 11G35
Mathematics Subject Classification ID: 11D61
zbMATH DE Number: 1375115
zbMATH Keywords: Diophantine equations over function fields
reviewed by: Jan-Hendrik Evertse
MaRDI profile type: Publication
full work available at URL: jnth.1999.2400
OpenAlex ID: W2092247402
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DOI: 10.1006/JNTH.1999.2400
description: scientific article; zbMATH DE number 1375115