Scalar polynomial functions on the n× n matrices over a finite field

Abstract

Let F= GF( q) denote the finite field of order q, and let ƒ(x)ϵF[x]. Then f( x) defines, via substitution, a function from F n× n , the n× n matrices over F, to itself. Any function ƒ:F n×n → F n×n which can be represented by a polynomial f( x) ϵF[ x] is called a scalar polynomial function on F n× n . After first determining the number of scalar polynomial functions on F n× n , the authors find necessary and sufficient conditions on a polynomial ƒ(x) ϵ F[x] in order that it defines a permutation of (i) D n , the diagonalizable matrices in F n× n , (ii) R n , the matrices in F n× n all of whose roots are in F, and (iii) the matric ring F n× n itself. The results for (i) and (ii) are valid for an arbitrary field F.