On growth of the number of determinants with restricted entries

TL;DR

This paper investigates how the number of distinct determinants of matrices with entries from a finite set grows as the matrix size increases, revealing new bounds and growth patterns.

Contribution

It provides new bounds and insights into the growth of the set of determinants for matrices with restricted entries, extending previous understanding.

Findings

Derived bounds on the size of determinant sets for matrices over finite sets

Established growth rates of determinant sets as matrix dimension increases

Connected determinant set size to properties of the underlying set A

Abstract

Let $A$ be a finite subset of a field $\mathbb{F}$ and $D_n(A)$ be a set of all matrices with entries in $A$, namely $$ D_n(A)=\{D\in \mathbb{F}\ |\ \exists a_{ij}\in A, 1 \le i,j \le n, \det\bigl((a_{ij})\bigr)=D\}, $$ where the symbol $(a_{ij})$ defines the matrix with elements $a_{ij}$. How big is the size of the set $D_n(A)$ comparing to the size of the set $A$?