Linearized polynomial maps over finite fields

TL;DR

This paper explores the structure of linearized polynomial maps over finite fields, revealing their correspondence with matrices over polynomial rings and solving longstanding conjectures like the Jacobian Conjecture for these maps.

Contribution

It establishes a novel matrix-based framework for linearized polynomial automorphisms over finite fields, enabling solutions to famous conjectures.

Findings

Linearized polynomial maps correspond to matrices over polynomial rings.

Composition of maps translates to matrix multiplication.

The framework solves the Jacobian Conjecture for these polynomials.

Abstract

We consider polynomial maps described by so-called "(multivariate) linearized polynomials". These polynomials are defined using a fixed prime power, say q. Linearized polynomials have no mixed terms. Considering invertible polynomial maps without mixed terms over a characteristic zero field, we will only obtain (up to a linear transformation of the variables) triangular maps, which are the most basic examples of polynomial automorphisms. However, over the finite field F_q automorphisms defined by linearized polynomials have (in general) an entirely different structure. Namely, we will show that the linearized polynomial maps over F_q are in one-to-one correspondence with matrices having coefficients in a univariate polynomial ring over F_q. Furthermore, composition of polynomial maps translates to matrix multiplication, implying that invertible linearized polynomial maps correspond to invertible matrices. This alternate description of the linearized polynomial automorphism subgroup leads to the solution of many famous conjectures (most notably, the Jacobian Conjecture) for this kind of polynomials and polynomial maps.