{"title":"有限域上齐次多项式方程组解个数的组合方法","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":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":"{\"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\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2018-07-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"4\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.17323/1609-4514-2022-22-4-565-593\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.17323/1609-4514-2022-22-4-565-593","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
A Combinatorial Approach to the Number of Solutions of Systems of Homogeneous Polynomial Equations over Finite Fields
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
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