A probabilistic approach to value sets of polynomials over finite fields

TL;DR

This paper investigates the distribution of value set sizes for random polynomials over finite fields, deriving exact probabilities and demonstrating normal distribution tendencies as the field size grows.

Contribution

It provides the first exact probability distribution for value set sizes and links these results to cyclotomic mappings, advancing understanding of polynomial value distributions over finite fields.

Findings

Number of missing values tends to a normal distribution as q increases

Derived exact probability distribution for value set sizes

Analyzed a variation involving union of random sets

Abstract

In this paper we study the distribution of the size of the value set for a random polynomial with degree at most $q-1$ over a finite field $\mathbb{F}_q$. We obtain the exact probability distribution and show that the number of missing values tends to a normal distribution as $q$ goes to infinity. We obtain these results through a study of a random $r$-th order cyclotomic mappings. A variation on the size of the union of some random sets is also considered.