Distribution of the number of zeros of polynomials over a finite field

TL;DR

This paper investigates how the number of zeros of multivariable polynomials over finite fields is distributed, deriving probability generating functions and showing convergence to a Poisson distribution in the single-variable case.

Contribution

It provides a new analysis of the distribution of zeros for polynomials over finite fields, including explicit generating functions and asymptotic behavior.

Findings

Distribution converges to Poisson as degree and field size grow

Explicit probability generating functions derived for bounded degree polynomials

Distribution characterized for multivariable cases

Abstract

We study the probability distribution of the number of zeros of multivariable polynomials with bounded degree over a finite field. We find the probability generating function for each set of bounded degree polynomials. In particular, in the single variable case, we show that as the degree of the polynomials and the order of the field simultaneously approach infinity, the distribution converges to a Poisson distribution.