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

TL;DR

This work analyzes the distribution and expectation of the number of common zeros in random polynomial systems over finite fields and rings. By introducing structural sample spaces that either contain all functions or extend a ring, the authors prove that the zero-count $N$ is binomial with parameters $(q^n, 1/q^m)$ when sampling from spaces containing functions, and they compute the exact expectation $|R|^{n-m}$ for sampling from an $R$-module extending $R$ over finite rings. The results unify and extend prior work by treating broad classes of sample spaces and by transitioning to ring settings via density arguments for infinite spaces, plus asymptotic Poisson behavior in certain limits. These findings have theoretical significance for random polynomial systems and potential cryptographic applications, and they open avenues for further geometric and asymptotic investigations of random varieties over finite fields and rings.

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.

Theorems & Definitions (28)

• Definition 1.1
• Definition 1.2
• Theorem 1
• Theorem 2
• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Lemma 3.3
• proof