  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.
