Asymptotic distribution of the number of zeros in random polynomials in a given equivalence class over a finite field

TL;DR

This paper investigates the distribution of zeros in random polynomials over finite fields within a specific equivalence class, showing asymptotic Poisson behavior and extending previous results with implications for Reed-Solomon codes.

Contribution

It establishes the asymptotic Poisson distribution of zeros for polynomials in Hayes equivalence classes and provides formulas for their counts under certain conditions, extending earlier work.

Findings

Zeros follow a Poisson distribution asymptotically.

Formulas for polynomial counts when degree is proportional to q.

Supports the Deep-Hole Conjecture for Reed-Solomon codes.

Abstract

Hayes equivalence is defined on monic polynomials over a finite field $\fq$ in terms of the prescribed leading coefficients and the residue classes modulo a given monic polynomial $Q$. We study the distribution of the number of zeros in a random polynomial over finite fields in a given Hayes equivalence class. It is well known that the number of distinct zeros of a random polynomial over $\fq$ is asymptotically Poisson with mean 1. We show that this is also true for random polynomials in any given Hayes equivalence class. Asymptotic formulas are also given for the number of such polynomials when the degree of such polynomials is proportional to $q$ and the degree of $Q$ and the number of prescribed leading coefficients are bounded by $\sqrt{q}$. When $Q=1$, the problem is equivalent to the study of the distance distribution in Reed-Solomon codes. Our asymptotic formulas extend some earlier results and imply that all words for a large family of Reed-Solomon codes are ordinary, which further supports the well-known {\em Deep-Hole} Conjecture.