A Dynamical Lifting Problem For Additive Polynomials

TL;DR

This paper explores a dynamical analogue of the lifting problem for Galois covers, providing a negative result for additive polynomials over algebraic closures of finite fields and computing related conjugacy class dimensions.

Contribution

It introduces a new dynamical lifting problem framework and explicitly analyzes additive, separable polynomials over algebraic closures of finite fields.

Findings

Negative solution for the dynamical lifting problem for additive polynomials

Explicit computation of the dimension of conjugacy classes containing additive polynomials

Abstract

We introduce a dynamical analogue of the lifting problem for Galois covers of algebraic curves and find a negative solution for the collection of additive, separable polynomials over $\overline{\mathbb{F}}_p$. We also explicitly compute the dimension of the space of linear conjugacy classes in $M_{p^m}(\overline{\mathbb{F}}_p)$ which contain an additive, separable polynomial.