{"title":"A Combinatorial Approach to the Number of Solutions of Systems of Homogeneous Polynomial Equations over Finite Fields","authors":"Peter Beelen, M. Datta, S. Ghorpade","doi":"10.17323/1609-4514-2022-22-4-565-593","DOIUrl":null,"url":null,"abstract":"We give a complete conjectural formula for the number $e_r(d,m)$ of maximum possible ${\\mathbb{F}}q$-rational points on a projective algebraic variety defined by $r$ linearly independent homogeneous polynomial equations of degree $d$ in $m+1$ variables with coefficients in the finite field ${\\mathbb{F}}q$ with $q$ elements, when $d<q$. It is shown that this formula holds in the affirmative for several values of $r$. In the general case, we give explicit lower and upper bounds for $e_r(d,m)$ and show that they are sometimes attained. Our approach uses a relatively recent result, called the projective footprint bound, together with results from extremal combinatorics such as the Clements-Lindstr\\\"om Theorem and its variants. Applications to the problem of determining the generalized Hamming weights of projective Reed-Muller codes are also included.","PeriodicalId":54736,"journal":{"name":"Moscow Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2018-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Moscow Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.17323/1609-4514-2022-22-4-565-593","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 4
Abstract
We give a complete conjectural formula for the number $e_r(d,m)$ of maximum possible ${\mathbb{F}}q$-rational points on a projective algebraic variety defined by $r$ linearly independent homogeneous polynomial equations of degree $d$ in $m+1$ variables with coefficients in the finite field ${\mathbb{F}}q$ with $q$ elements, when $d
期刊介绍：
The Moscow Mathematical Journal (MMJ) is an international quarterly published (paper and electronic) by the Independent University of Moscow and the department of mathematics of the Higher School of Economics, and distributed by the American Mathematical Society. MMJ presents highest quality research and research-expository papers in mathematics from all over the world. Its purpose is to bring together different branches of our science and to achieve the broadest possible outlook on mathematics, characteristic of the Moscow mathematical school in general and of the Independent University of Moscow in particular.
An important specific trait of the journal is that it especially encourages research-expository papers, which must contain new important results and include detailed introductions, placing the achievements in the context of other studies and explaining the motivation behind the research. The aim is to make the articles — at least the formulation of the main results and their significance — understandable to a wide mathematical audience rather than to a narrow class of specialists.
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