Counting points in a small box on varieties
type: Item
instance of: scholarly article
title: Counting points in a small box on varieties
zbMATH Open document ID: 0669.10045
author: Masahiko Fujiwara
published in: Proceedings of the Japan Academy. Series A
publication date: 1988-01-01T00:00:00Z
review text: The number of solutions of bounded size (or more generally, in a box) of a system of congruences \[ G_ 1(x)\equiv G_ 2(x)\equiv...\equiv G_ s(x)\equiv 0 mod p \] (G\({}_ i\) homogeneous polynomials with integral coefficients in n variables) is estimated in both directions, showing that the solutions are uniformly distributed. The validity of the estimate depends on p and the box being large enough, but does not need n to be large [this was the case in previous results of \textit{W. M. Schmidt}, Diophantine Analysis, Lond. Math. Soc. Lect. Note Ser. 109, 37- 66 (1986; Zbl 0589.10024)]. It also depends on a nonsingularity condition for the polynomials in the pencil of \(G_ 1,...,G_ s\). This condition, which previously was assumed to be natural, is shown to be true only rarely if \(s\geq 3.\) The paper gives only an outline of the proofs, postponing details to a later article (see the following review).
Mathematics Subject Classification ID: 11E76
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11L03
Mathematics Subject Classification ID: 11D72
zbMATH DE Number: 4095582
zbMATH Keywords: varieties
zbMATH Keywords: finite fields
zbMATH Keywords: small solutions of congruences
zbMATH Keywords: systems of forms
zbMATH Keywords: points in boxes
zbMATH Keywords: system of congruences
reviewed by: Rainer Schulze-Pillot
MaRDI profile type: Publication
full work available at URL: pjaa.64.267
OpenAlex ID: W1975376693
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DOI: 10.3792/PJAA.64.267
description: scientific article; zbMATH DE number 4095582