Word maps, polynomial maps and image ratios

TL;DR

This paper studies the distribution of image ratios of word and polynomial maps on finite groups and rings, showing that these ratios can be dense in the interval [0,1], revealing complex behavior of such maps.

Contribution

It demonstrates the existence of word and polynomial maps with image ratios dense in [0,1] for finite groups and rings, expanding understanding of their possible behaviors.

Findings

Existence of maps with dense image ratios in [0,1]

Analysis of image ratios on finite groups and rings

Extension of results to polynomial maps in rings

Abstract

If $A$ is a finite group (or a finite ring) and $\omega$ is a word map (or a polynomial map), we define the quantity $|\omega(A)|/|A|$ as the image ratio of $\omega$ on $A$ and will be denoted by $\mu(\omega,A)$. In this article, we investigate the set $\mathrm{R}(\omega)=\{\mu(\omega,A) : A \text{ is a finite group}\}$, and also consider the case of rings. Specifically, we demonstrate the existence of word maps (and polynomial maps) whose set of image ratios is dense in $[0,1]$ for both groups (and rings).