Improvements upon the Chevalley-Warning-Ax-Katz-type estimates
type: Item
instance of: scholarly article
title: Improvements upon the Chevalley-Warning-Ax-Katz-type estimates
zbMATH Open document ID: 1201.11004
author: Wei Cao
author: Qi Sun
published in: Journal of Number Theory
publication date: 2007-02-12T00:00:00Z
review text: Let \(f\) be a polynomial over a finite field \(\mathbb F_q\) with \(q\) elements and let \(N(f=0)\) denote the number of zeros of \(f\) in \(\mathbb F_q\). Let \(e_p\) denote the exponent of \(p\) in \(N(f=0)\). The Chevalley-Warning, Ax-Katz and Moreno-Moreno theorems give non-trivial lower bounds for \(e_p\) of increasing strength. Given \(f\) the authors define two associated polynomials \(\hat f\) and \(\skew6\hat{\hat f}\) that are possibly of lower degree than \(f\), but have the same number of zeros. On applying the above classical results with \(\hat f\) or \(\skew6\hat{\hat f}\) instead of \(f\), one then sometimes obtains a larger lower bound for \(e_p\). In case of \(\hat f\) the proof that \(N(f=0)=N({\hat f}=0)\) is nice, short and elementary. For \(\skew6\hat{\hat f}\) it relies on results from the earlier paper [Finite Fields Appl. 12, 681--692 (2006; Zbl 1169.11055)] by the same authors.
reviewed by: Pieter Moree
Mathematics Subject Classification ID: 11A07
Mathematics Subject Classification ID: 11D72
Mathematics Subject Classification ID: 11T06
zbMATH DE Number: 5124510
zbMATH Keywords: solutions of equations
zbMATH Keywords: finite field
zbMATH Keywords: \(p\)-adic reduction
MaRDI profile type: Publication
full work available at URL: j.jnt.2006.04.003
OpenAlex ID: W2054196558
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DOI: 10.1016/J.JNT.2006.04.003
description: scientific article; zbMATH DE number 5124510