On the determinants and permanents of matrices with restricted entries over prime fields

TL;DR

This paper investigates the number of distinct determinants and permanents of matrices with entries from a subset of a prime field, providing bounds when the subset size is small.

Contribution

It establishes new lower bounds on the number of distinct determinants and permanents for matrices over prime fields with restricted entries.

Findings

Almost |A|^{3+1/45} distinct determinants

Almost |A|^{2-1/6} distinct permanents

Results hold when |A| is sufficiently small

Abstract

Let $A$ be a set in a prime field $\mathbb{F}_p$. In this paper, we prove that $d\times d$ matrices with entries in $A$ determine almost $|A|^{3+\frac{1}{45}}$ distinct determinants and almost $|A|^{2-\frac{1}{6}}$ distinct permanents when $|A|$ is small enough.