A Schwartz-Zippel Type Estimate for Homogenous Finite Field Polynomials

TL;DR

This paper develops a probabilistic recursion method to estimate the number of zeros of homogeneous polynomials over finite fields, providing bounds and applications in graph theory and polynomial collections.

Contribution

It introduces a Schwartz-Zippel type estimate specifically for homogeneous finite field polynomials using a novel probabilistic recursion approach.

Findings

Established upper and lower bounds for zeros of homogeneous polynomials

Applied results to perfect matching problems in bipartite graphs

Analyzed common zeros in collections of polynomials

Abstract

In this paper, we obtain a Schwartz-Zippel type estimate for homogenous finite field polynomials. Specifically, we use a probabilistic recursion technique to find upper and lower bounds for the number of zeros of a homogenous polynomial and illustrate our result with two examples involving perfect matching in bipartite graphs and common zeros in a collection of polynomials, respectively.