Classification of affine operators up to biregular conjugacy

TL;DR

This paper characterizes when two affine operators are biregularly conjugate over an algebraically closed field of characteristic zero, providing necessary and sufficient conditions and a canonical form for classification.

Contribution

It extends previous work by establishing a complete classification of affine operators up to biregular conjugacy with explicit criteria and canonical forms.

Findings

Necessary and sufficient conditions for biregular conjugacy.

Canonical form of affine operators under biregular conjugacy.

Extension of classification results to non-bijective affine operators.

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].