A note on the proof of a theorem of Katz
type: Item
instance of: scholarly article
title: A note on the proof of a theorem of Katz
zbMATH Open document ID: 1144.11082
author: Xiang-Dong Hou
published in: Finite Fields and their Applications
publication date: 2005-06-01T00:00:00Z
review text: Let \(f_i\in\mathbb F_q[X_1,\dots,X_n]\) be polynomials of degree \(d_i\), \(1\leq i\leq r\), where \(d_1\geq\cdots\geq d_r\geq 1\). Denote the set of zeros of \(f_i\) in \(\mathbb F_q^n\) by \(Z(f_i)\). \textit{N. M. Katz} [Am. J. Math. 93, 485--499 (1971; Zbl 0237.12012)] proved that \(q^{\lceil\frac {n-d_1-\cdots-d_r}{d_1}\rceil}\) divides \(|Z(f_1)\cap\cdots\cap Z(f_r)|\). A more elementary proof of this result was given by \textit{D. Wan} [Am. J. Math. 111, 1--8 (1989; Zbl 0683.12010)]. Here the author gives a new and simpler proof of this result.
reviewed by: Olaf Ninnemann
Mathematics Subject Classification ID: 11T24
Mathematics Subject Classification ID: 11L40
Mathematics Subject Classification ID: 11T30
zbMATH DE Number: 2173081
zbMATH Keywords: finite field
zbMATH Keywords: polynomials
zbMATH Keywords: \(p\)-adic bound
MaRDI profile type: Publication
full work available at URL: j.ffa.2004.11.002
OpenAlex ID: W2079156824
cites work: Zeroes of Polynomials Over Finite Fields
cites work: On a Theorem of Ax
cites work: An Elementary Proof of a Theorem of Katz
cites work: A Chevalley-Warning Approach To p-Adic Estimates of Character Sums
DOI: 10.1016/J.FFA.2004.11.002
description: scientific article; zbMATH DE number 2173081