Strongly nilpotent automorphisms are Pascal finite

Abstract: We compare two classes of polynomial automorphisms, strongly nilpotent and Pascal finite. We conclude that every strongly nilpotent automorphism is a Pascal finite one, but not vice versa. We observe that Nagata's automorphism is Pascal finite, but not strongly nilpotent. Considering Vasyunin example leads us to conclusion that not every quadratic polynomial automorphism is Pascal finite.

• The paper establishes that strongly nilpotent automorphisms are a subset of Pascal finite ones, linked via linear triangularizability and sharp degree bounds.
• It provides necessary and sufficient criteria based on the nilpotency of affine linear parts and iterative derivation operators for identifying Pascal finiteness.
• Counterexamples like the Nagata automorphism reveal that Pascal finiteness does not imply strong nilpotence, highlighting distinct algebraic behaviors.

Strongly Nilpotent Automorphisms and Pascal Finiteness

Introduction

This paper addresses the structural and algorithmic interplay between strongly nilpotent and Pascal finite polynomial automorphisms, with primary focus on the context of the Jacobian conjecture. The authors rigorously establish that the class of strongly nilpotent automorphisms is strictly included in the class of Pascal finite ones, deliver necessary and sufficient criteria for Pascal finiteness based on nilpotency, and demonstrate that certain classical automorphisms, such as the Nagata automorphism, are Pascal finite without being strongly nilpotent. The analysis is extended with explicit counterexamples and connections to triangularizability, the theory of tame automorphisms, and combinatorial inversion schemes.

Background: Polynomial Automorphisms, the Jacobian Conjecture, and Finiteness Conditions

Let $K$ be a field (typically of characteristic zero) and $K[X]=K[X_1,\ldots,X_n]$ the polynomial ring in $n$ variables. A polynomial automorphism is an invertible polynomial map $F: K^n \to K^n$. The Jacobian conjecture posits that any polynomial mapping $F$ with constant invertible Jacobian determinant admits a polynomial inverse.

Key automorphism notions:

• Keller maps: polynomial endomorphisms with constant, invertible Jacobian.
• Triangular, affine, tame automorphisms: various subgroups of the automorphism group, characterized by their algebraic structure and generating sets.

The problem of finiteness for automorphism classes is addressed via the notion of locally finite endomorphisms (those for which orbits under iteration are finite-dimensional in the appropriate sense) and the more restrictive “Pascal finiteness” property, which is algorithmically motivated by the recursive structure of the inversion formula for polynomial maps.

Pascal Finiteness: Definition and Characterization

A polynomial automorphism $F$ is said to be Pascal finite if repeated application of a derivation-like operator $\Delta_F: K[X]^n \to K[X]^n$ (defined by $\Delta_F(P)=P \circ F - P$, iterated $m$ times on the identity map) yields zero: $\Delta_F^m(\text{Id}) = 0$ for some $K[X]=K[X_1,\ldots,X_n]$0. Equivalently, $K[X]=K[X_1,\ldots,X_n]$1 is a root of $K[X]=K[X_1,\ldots,X_n]$2 in the context of this derivation algebra. The paper formally characterizes Pascal finiteness:

• It is invariant under conjugation by automorphisms.
• All triangular and linearly triangularizable automorphisms are Pascal finite.
• The class is closed under inversion and iteration, but not composition.
• Pascal finitely automorphisms are a strict subclass of locally finite ones.

A pivotal structural result is that for affine maps of the form $K[X]=K[X_1,\ldots,X_n]$3, Pascal finiteness is equivalent to nilpotency of $K[X]=K[X_1,\ldots,X_n]$4. More generally, cubic polynomial maps with low nilpotence indices are shown to be Pascal finite.

Strongly Nilpotent Automorphisms: Structure and Implications

A polynomial map $K[X]=K[X_1,\ldots,X_n]$5 is strongly nilpotent if the Jacobian matrix $K[X]=K[X_1,\ldots,X_n]$6 is strongly nilpotent: products of its values at arbitrary vectors, of length $K[X]=K[X_1,\ldots,X_n]$7 for some $K[X]=K[X_1,\ldots,X_n]$8, are always zero. The paper assembles the following equivalences:

• $K[X]=K[X_1,\ldots,X_n]$9 is strongly nilpotent if and only if $n$0 is linearly triangularizable.
• In such cases, $n$1, a sharp bound superior to the general $n$2 estimate.

This triangulability is central to the main theorem linking strong nilpotence and Pascal finiteness.

Main Result and Contrasting Examples

The core theorem established is:

Every strongly nilpotent automorphism is Pascal finite. However, not all Pascal finite automorphisms are strongly nilpotent.

This is shown by observing that strong nilpotence implies linearly triangularizability, which, per earlier results, implies Pascal finiteness.

The strictness of the inclusion is explicitly demonstrated via the Nagata automorphism: it is Pascal finite but not strongly nilpotent nor tame. The Vasyunin example shows that invertible quadratic automorphisms need not be Pascal finite—this is a contradictory claim to naive expectations and highlights the limitations of Pascal finiteness as a universal criterion in low degree.

Relation with Tame Automorphisms

The work carefully distinguishes Pascal finite and tame automorphisms:

• The two classes intersect (e.g., triangular and elementary automorphisms are both), but neither contains the other.
• Affine automorphisms need not be Pascal finite unless their linear part is nilpotent.
• The Nagata automorphism is a central example of Pascal finite but wild (non-tame) automorphism.

Combinatorial and Theoretical Implications

The authors draw connections between Pascal finiteness and combinatorial methods in the study of polynomial involutions and inverses, notably the weighted rooted tree enumerations of inverse coefficients. They observe that the operator $n$3 effectively generates the relevant combinatorial classes, making Pascal finiteness a natural algebraic closure for results arising from the Shuffling Conjecture and related combinatorics. This points to analytical opportunities for investigating Pascal finite automorphisms outside the scope of strong nilpotence, especially by employing planar trees and graph invariants.

Conclusion

The strict inclusion of strongly nilpotent automorphisms inside the Pascal finite class provides an algorithmic and structural bridge between classical triangulability and inversion protocols in polynomial automorphism theory. The distinction is illuminated through explicit examples, and the framework is tied to deep combinatorial methods and degree bounds. The results clarify the limitations of Pascal finiteness and suggest further avenues in the combinatorial analysis of automorphism classes not covered by strong nilpotence—for instance, studying the algebraic and dynamic properties of automorphisms like the Nagata and Vasyunin examples, and relating their inversion complexity to rooted tree expansions and beyond.

Future directions include extending combinatorial techniques to the broader Pascal finite context and exploring potential implications for the Jacobian conjecture, especially regarding the existence and classification of exceptional invertible mappings in higher dimensions.