Linearized polynomial maps over finite fields

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 Fq automorphisms defined by linearized polynomials have (in general) an entirely different structure. Namely, we will show that the linearized polynomial maps over Fq are in one-to-one correspondence with matrices having coefficients in a univariate polynomial ring over Fq. 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.