Geometry of word equations in simple algebraic groups over special fields

TL;DR

This paper surveys recent advances in understanding word equations in simple algebraic groups and matrix algebras, including new results on the images of word maps over various special fields.

Contribution

It provides a comprehensive overview of recent developments and introduces new findings on the behavior of word maps over complex, real, p-adic, and finite fields.

Findings

New results on the image of word maps over special fields

Survey of recent progress in algebraic group word equations

Analysis of polynomial equations in matrix algebras

Abstract

This paper contains a survey of recent developments in investigation of word equations in simple matrix groups and polynomial equations in simple (associative and Lie) matrix algebras along with some new results on the image of word maps on algebraic groups defined over special fields: complex, real, p-adic (or close to such), or finite.