The number of solutions of a random system of polynomials over a finite field

TL;DR

This paper investigates the distribution of common zeros of random polynomial systems over finite rings and fields, providing expected values and conditions for binomial distribution in the case of fields.

Contribution

It computes the expected number of solutions over finite rings and establishes binomial distribution conditions for solutions over fields.

Findings

Expected number of solutions over finite rings calculated

Number of solutions over fields follows a binomial distribution under certain conditions

Provides necessary and sufficient conditions for the distribution of solutions

Abstract

We study the probability distribution of the number of common zeros of a system of $m$ random $n$-variate polynomials over a finite commutative ring $R$. We compute the expected number of common zeros of a system of polynomials over $R$. Then, in the case that $R$ is a field, under a necessary-and-sufficient condition on the sample space, we show that the number of common zeros is binomially distributed.