On rational fixed points of finite group actions on the affine space

TL;DR

This paper proves the existence of rational fixed points for finite group actions on affine spaces under specific conditions related to the field's properties and dimension.

Contribution

It establishes new conditions guaranteeing rational fixed points for finite group actions on affine spaces, extending previous results in algebraic geometry.

Findings

Existence of rational fixed points when k is p-special and p ≠ char(k)

Existence of rational fixed points when k is perfect, fertile, and n=3

Applicable to finite l-group actions on affine n-space

Abstract

Consider a finite l-group acting on the affine space of dimension n over a field k, whose characteristic differs from l. We prove the existence of a fixed point, rational over k, in the following cases: --- The field k is p-special for some prime p different from its characteristic. --- The field k is perfect and fertile, and n = 3.