Smith normal form of augmented degree matrix and its applications
type: Item
instance of: scholarly article
title: Smith normal form of augmented degree matrix and its applications
zbMATH Open document ID: 1184.15012
author: Wei Cao
published in: Linear Algebra and its Applications
publication date: 2009-10-09T00:00:00Z
review text: Let \(\mathbb{F}_{q}\) be a finite field of \(q\) elements and \(p\) its characteristic. Let \(f\left( x_{1},x_{2},\dots,x_{n}\right) \) be a nonzero polynomial in \(n\) variables over \(\mathbb{F}_{q}\). \ For each positive integer \(r\), let \(N_{q}(f)\) denote the number of \(\mathbb{F}_{q^{r}}\)-rational points on the affine hypersurface \(f=0\), i.e. \[ N_{q}(f)=\#\left\{ \left( x_{1},x_{2},\dots,x_{n}\right) \in \mathbb{A} ^{n}\left( \mathbb{F}_{q^{r}}\right) \left| f\left( x_{1},x_{2},\dots,x_{n}\right) =0\right. \right\} \] The author uses the Smith normal forms of an augmented degree matrix to evaluate the number \(N_{q}(f).\) This work is a generalization of the author's previous results. As he mentions, one of the benefit of this result is that there are algorithms for finding the Smith normal form of a matrix, and these are programmed into common Computer Algebra packages such as Maple and Mupad. He gives, at the end of the paper an example to compare this result with the results previously obtained.
reviewed by: Manouchehr Misaghian
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 11T24
Mathematics Subject Classification ID: 11C20
Mathematics Subject Classification ID: 15B33
zbMATH DE Number: 5612555
zbMATH Keywords: finite field
zbMATH Keywords: polynomial
zbMATH Keywords: augmented degree matrix
zbMATH Keywords: Smith normal form
zbMATH Keywords: Gauss sum
describes a project that uses: GAP
describes a project that uses: Maple
describes a project that uses: EDIM
MaRDI profile type: Publication
full work available at URL: j.laa.2009.06.018
OpenAlex ID: W2010070603
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DOI: 10.1016/J.LAA.2009.06.018
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description: scientific article; zbMATH DE number 5612555