Classification of affine operators up to biregular conjugacy

Abstract

Let f(x)=Ax+b and g(x)=Cx+d be two affine operators given by n-by-n matrices A and C and vectors b and d over a field F. They are said to be biregularly conjugate if hf=gh for some bijection h: F^n-->F^n being biregular, this means that the coordinate functions of h and h^{-1} are polynomials. Over an algebraically closed field of characteristic 0, we obtain necessary and sufficient conditions of biregular conjugacy of affine operators and give a canonical form of an affine operator up to biregular conjugacy. These results for bijective affine operators were obtained by J.Blanc [Conjugacy classes of affine automorphisms of K^n and linear automorphisms of P^n in the Cremona groups, Manuscripta Math. 119 (2006) 225-241].