# On the Distribution of Discriminants over a Finite Field

## TL;DR

This paper investigates the distribution of discriminants of monic polynomials over finite fields, establishing conditions under which they are evenly distributed and providing a converse theorem for certain cases.

## Contribution

It proves conditions for the equal distribution of polynomial discriminants over finite fields and offers a converse theorem for squarefree cases.

## Key findings

- Discriminants are equally distributed under specific gcd conditions.
- Equal distribution depends on the parity of the finite field size.
- A converse theorem is established for squarefree cases.

## Abstract

For a prime power $q$, we show that the discriminants of monic polynomials in $\mathbb{F}_q[x]$ of a fixed degree $m$ are equally distributed if $\gcd(q-1,m(m-1))=2$ when $q$ is odd and $\gcd(q-1,m(m-1))=1$ if $q$ is even. A theorem in the converse direction is proved when $q-1$ is squarefree.