On the number of rational points on codimension 1 algebraic sets in \(\mathbb{P}^ n (\mathbb{F}_ q)\)
type: Item
instance of: scholarly article
title: On the number of rational points on codimension 1 algebraic sets in \(\mathbb{P}^ n (\mathbb{F}_ q)\)
zbMATH Open document ID: 0816.14009
DOI: 10.1016/0012-365X(93)E0009-S
author: Anders Bjært Sørensen
published in: Discrete Mathematics
publication date: 1995-02-01T00:00:00Z
review text: As an example of the type of results obtained by the author, we have the following result. Let \(X \subset \mathbb{P}^ n (\mathbb{F}_ q)\) be a pure \((n - 1)\)-dimensional algebraic set which is defined over \(\mathbb{F}_ q\). Suppose \(X\) has degree \(d\) and let \(| X |_{\mathbb{F}_ q}\) denote the number of points of \(X\) rational over \(\mathbb{F}_ q\). Theorem: We have \(| X |_{\mathbb{F}_ q} \leq dq^{n-1} + {q^{n- 1} - 1 \over q - 1}\). These results are then generalized through the use of coding theory, and applications to coding theory are given.
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14N10
Mathematics Subject Classification ID: 94B27
Mathematics Subject Classification ID: 14G15
zbMATH DE Number: 716466
zbMATH Keywords: Weil conjectures
zbMATH Keywords: finite ground field
zbMATH Keywords: number of rational points
zbMATH Keywords: coding theory
reviewed by: David Goss
MaRDI profile type: Publication
cites work: Q4081121
cites work: On generalized ReedMuller codes and their relatives
cites work: Q3851217
cites work: Q3324075
cites work: Projective Reed-Muller codes
cites work: Q5795562
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description: scientific article; zbMATH DE number 716466