A Combinatorial Approach to the Number of Solutions of Systems of Homogeneous Polynomial Equations over Finite Fields
Beelen, Peter and Datta, Mrinmoy and Ghorpade, Sudhir R. (2022) A Combinatorial Approach to the Number of Solutions of Systems of Homogeneous Polynomial Equations over Finite Fields. Moscow Mathematical Journal, 22 (4). pp. 565593. ISSN 16094514
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Abstract
We give a complete conjectural formula for the number er(d, m) of maximum possible Fqrational 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 Fq 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 er(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–Lindström Theorem and its variants. Applications to the problem of determining the generalized Hamming weights of projective Reed–Muller codes are also included. © 2022 Independent University of Moscow.
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Item Type:  Article  
Additional Information:  Peter Beelen gratefully acknowledges the support from The Danish Council of Scientific Research (DFFFNU) for the project Correcting on a Curve, Grant No. 802100030B.  
Uncontrolled Keywords:  Finite field; footprint bound; generalzed Hamming weight; projective algebraic variety; projective Reed–Muller code  
Subjects:  Mathematics Mathematics > Numerical analysis 

Divisions:  Department of Mathematics  
Depositing User:  . LibTrainee 2021  
Date Deposited:  23 Nov 2022 11:53  
Last Modified:  23 Nov 2022 11:53  
URI:  http://raiith.iith.ac.in/id/eprint/11397  
Publisher URL:  https://doi.org/10.17323/1609451420222245655...  
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