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. 565-593. ISSN 1609-4514

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Abstract

We give a complete conjectural formula for the number er(d, m) of maximum possible Fq-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 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 com-binatorics 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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IITH Creators:

IITH Creators	ORCiD
Datta, Mrinmoy	UNSPECIFIED

Item Type:

Article

Additional Information:

Peter Beelen gratefully acknowledges the support from The Danish Council of Scientific Research (DFF-FNU) for the project Correcting on a Curve, Grant No. 8021-00030B.

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