Noncommutative polynomial maps

TL;DR

This paper explores polynomial maps in Ore extensions, emphasizing pseudo-linear transformations and their applications, including polynomial factorization over finite fields with Frobenius automorphism.

Contribution

It introduces a framework for polynomial maps in Ore extensions and connects factorizations in Ore extensions to those in classical polynomial rings.

Findings

Factorizations in Ore extensions over finite fields can be translated to classical polynomial factorizations.

Pseudo-linear transformations are crucial tools in understanding polynomial maps in Ore extensions.

The paper provides methods to factor polynomials in Ore extensions using known techniques from polynomial rings.

Abstract

Polynomial maps attached to polynomials of an Ore extension are naturally defi ned. In this setting we show the importance of pseudo-linear transformations and give some applications. In particular, factorizations of polynomials in an Ore extension over a fi nite fi eld F_q[t;S ], where S is the Frobenius automorphism, are translated into factorizations in the usual polynomial ring F_q[x].