Vanishing Ideals of Projective Spaces over Finite Fields and a Projective Footprint Bound  

作　　者：Peter BEELEN Mrinmoy DATTA Sudhir R.GHORPADE 

机构地区：[1]Department of Applied Mathematics and Computer Science,Technical University of Denmark,DK 2800.Kgs.Lyngby,Denmark [2]Department of Applied Mathematics and Computer Science,Technical University of Denmark,DK 2800,Kgs.Lyngby,Denmark Current address:Department of Mathematics and Statistics,UiT-The Arctic Unwersity ofNorway,N-9037,Tromsφ,Norway [3]Department of Mathematics,Indian Institute of Technology Bombay,Powai,Mumbai 400076,India

出　　处：《Acta Mathematica Sinica,English Series》2019年第1期47-63,共17页数学学报（英文版）

基　　金：supported by the Danish Council for Independent Research(Grant No.DFF–4002-00367),supported by the Danish Council for Independent Research(Grant No.DFF–6108-00362);supported by the Research Council of Norway(Project No.280731);supported by IRCC Award grant 12IRAWD009 from IIT Bombay

摘　　要：We consider the vanishing ideal of a projective space over a finite field. An explicit set of generators for this ideal has been given by Mercier and Rolland. We show that these generators form a universal Gr¨obner basis of the ideal. Further we give a projective analogue for the so-called footprint bound, and a version of it that is suitable for estimating the number of rational points of projective algebraic varieties over finite fields. An application to Serre’s inequality for the number of points of projective hypersurfaces over finite fields is included.We consider the vanishing ideal of a projective space over a finite field. An explicit set of generators for this ideal has been given by Mercier and Rolland. We show that these generators form a universal Gr¨obner basis of the ideal. Further we give a projective analogue for the so-called footprint bound, and a version of it that is suitable for estimating the number of rational points of projective algebraic varieties over finite fields. An application to Serre’s inequality for the number of points of projective hypersurfaces over finite fields is included.

关 键 词：Finite field projective space algebraic variety vanishing ideal Grbner basis footprint bound projective hypersurface 

分 类 号：O1[理学—数学]