# Maximum Number of Common Zeros of Homogeneous Polynomials over Finite Fields

## TL;DR

This paper investigates the maximum number of common zeros of linearly independent homogeneous polynomials over finite fields, proving a new conjecture for several cases and settling it completely for degree three, with applications to algebraic geometry and coding theory.

## Contribution

It proves the validity of a new conjecture for the maximum number of zeros for various degrees, especially degree three, and extends results to specific cases like degree q-1 and q.

## Key findings

- Confirmed the new conjecture for several values of r.
- Settled the conjecture completely for degree d=3.
- Determined maximum zeros for d=q-1 and d=q cases.

## Abstract

About two decades ago, Tsfasman and Boguslavsky conjectured a formula for the maximum number of common zeros that $r$ linearly independent homogeneous polynomials of degree $d$ in $m+1$ variables with coefficients in a finite field with $q$ elements can have in the corresponding $m$-dimensional projective space. Recently, it has been shown by Datta and Ghorpade that this conjecture is valid if $r$ is at most $m+1$ and can be invalid otherwise. Moreover a new conjecture was proposed for many values of $r$ beyond $m+1$. In this paper, we prove that this new conjecture holds true for several values of $r$. In particular, this settles the new conjecture completely when $d=3$. Our result also includes the positive result of Datta and Ghorpade as a special case. Further, we determine the maximum number of zeros in certain cases not covered by the earlier conjectures and results, namely, the case of $d=q-1$ and of $d=q$. All these results are directly applicable to the determination of the maximum number of points on sections of Veronese varieties by linear subvarieties of a fixed dimension, and also the determination of generalized Hamming weights of projective Reed-Muller codes.