Classification of affine operators up to biregular conjugacy

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].

💡 Research Summary

The paper investigates the classification of affine operators on an n‑dimensional vector space over an algebraically closed field F of characteristic 0, up to biregular conjugacy. An affine operator is a map f : Fⁿ → Fⁿ of the form f(x)=Ax+b, where A∈Mₙ(F) is a linear part and b∈Fⁿ is a translation vector. Two such operators f and g are said to be biregularly conjugate if there exists a bijection h : Fⁿ → Fⁿ such that h and its inverse are both given by polynomial coordinate functions (i.e., h is a biregular automorphism of the affine space) and the intertwining relation h∘f = g∘h holds.

The authors first recall Blanc’s earlier work (Manuscripta Math. 119 (2006) 225‑241), which dealt with the special case where the affine operators are bijective (det A≠0). Blanc showed that, in the Cremona group, the conjugacy classes of affine automorphisms are determined by the Jordan normal form of the linear part and the position of the translation vector relative to the fixed‑point subspace. The present paper extends this analysis to arbitrary affine operators, allowing singular linear parts and translation vectors that may not lie in any fixed‑point subspace.

The core of the classification rests on two algebraic invariants that are preserved under any biregular conjugacy:

1. Spectrum and Jordan structure of the linear part. Because a biregular map is a polynomial automorphism, it cannot change the eigenvalues of the linear component, nor can it alter the sizes of the Jordan blocks associated with each eigenvalue. In particular, the multiset of eigenvalues (including multiplicities) of A must coincide with that of C, and for each eigenvalue λ the partition describing the dimensions of the Jordan blocks of A must equal the corresponding partition for C.

2. Nilpotent part. The nilpotent component (the blocks corresponding to eigenvalue 0) is especially rigid: any polynomial automorphism preserves the nilpotent index and the arrangement of the nilpotent Jordan blocks. Consequently, two affine operators can be biregularly conjugate only if their nilpotent parts are linearly conjugate.

Having fixed the linear part up to ordinary linear conjugacy, the remaining problem is to understand how the translation vectors b and d behave. The authors show that, once a linear isomorphism L∈GLₙ(F) satisfying C = LAL⁻¹ is chosen, the translation vectors must satisfy a compatibility condition of the form

  d = Lb + (I − C)Lv − Lv

for some vector v∈Fⁿ. Geometrically, this expresses that d is obtained from b by applying the same linear change of coordinates L and then adjusting by a “coboundary” term coming from the affine part of the conjugating map h(x)=Lx+v. If such an L and v exist, the map h(x)=Lx+v is itself biregular (its inverse is also affine), and the intertwining relation h∘f = g∘h holds. Conversely, any biregular conjugacy must arise in this way, which yields a complete necessary and sufficient condition for biregular conjugacy of arbitrary affine operators.

Armed with this criterion, the authors construct a canonical form for affine operators under biregular conjugacy. The procedure is:

• Bring the linear part A to its Jordan normal form, arranging blocks by eigenvalue.
• Within each Jordan block, use the freedom provided by the translation vector to simplify b as much as possible. The authors prove that, after an appropriate polynomial change of coordinates, all but at most one component of b can be eliminated. The remaining non‑zero component can be taken to be the first basis vector e₁ in the first Jordan block.
• The resulting canonical representative has the shape

  f_can(x) = diag(J_{k₁}(λ₁),…,J_{k_r}(λ_r))·x + e₁,

where each J_{k_i}(λ_i) is a Jordan block of size k_i with eigenvalue λ_i, and e₁ is the standard basis vector in the first block. All other blocks have zero translation part.

Two affine operators are biregularly conjugate if and only if their canonical forms coincide. This canonical form reduces to Blanc’s description when the linear part is invertible: in that case the translation vector must lie in the fixed‑point subspace, and the canonical form contains no non‑zero translation except possibly in the eigenvalue 1 block.

The paper concludes by discussing implications for the Cremona group and for the broader study of polynomial automorphisms of affine space. The authors note that the rigidity of the nilpotent part and the explicit compatibility condition for translation vectors provide a clear algebraic picture of how affine dynamics behave under polynomial change of coordinates. Potential applications include the classification of dynamical systems defined by affine maps, the study of invariant subvarieties under polynomial automorphisms, and computational approaches to detecting conjugacy in computer algebra systems.