An organizing principle in the study of the Jacobian Conjecture

Technical Summary: An Organizing Principle in the Study of the Jacobian Conjecture

Problem Statement
The paper addresses the Jacobian Conjecture (JC), which posits that every polynomial map $F: \mathbb{C}^n \to \mathbb{C}^n$ with a constant Jacobian determinant equal to 1 ($\det DF = 1$) is invertible (an automorphism). While the conjecture has been verified for specific classes of maps (e.g., quadratic maps), general proofs have historically failed. The author investigates the conjecture at the level of irreducible components of the algebraic locus $V^k_n$, defined as the set of polynomial maps with degree at most $k$ satisfying the Jacobian condition. The central question is whether the conjecture holds universally within a fixed irreducible component $\Omega \subset V^k_n$, or if it fails generically within that component.

Methodology
The paper employs a synthesis of algebraic geometry, complex analysis, and topology to establish a dichotomy regarding the invertibility of maps within an irreducible component $\Omega$.

1. Algebraic Formulation: The author defines the space of polynomial maps $P^k_n$ and the locus $V^k_n$. The set of invertible maps within this locus is denoted as $I^k_n = V^k_n \cap \text{Aut}(\mathbb{C}^n)$. The proof strategy relies on demonstrating that $I^k_n$ is an algebraic set (specifically, Zariski-closed) within $P^k_n$.
2. Constructibility and Chevalley's Theorem: To prove $I^k_n$ is algebraic, the author utilizes the isomorphism $\Lambda^k_n$ between coefficient space and polynomial maps. By analyzing the set of non-injective maps via the condition $\Lambda^k_n c(z) - \Lambda^k_n c(w) = 0$ with $z \neq w$, the author constructs a set $T$. Using Chevalley's theorem, the projection of this constructible set is shown to be constructible. Consequently, the set of injective maps (the complement of the projection) is also constructible.
3. Topological Closure (Analytic Argument): A critical step involves proving that the set of injective maps is not just constructible but Zariski-closed. The author argues that the Euclidean closure and Zariski closure of a constructible set coincide. Therefore, it suffices to prove the set is closed in the Euclidean topology.
   • A sequence of injective maps $F_j$ converging to a map $F$ is considered.
   • The Jacobian condition $\det DF_j = 1$ ensures $F$ is a local homeomorphism via the complex inverse function theorem.
   • A topological lemma (denoted as $(\dagger)$) is proven: If a sequence of injective local homeomorphisms converges locally uniformly to a local homeomorphism, the limit map is also injective. This relies on topological degree theory to show that if the limit map were non-injective, the approximating maps would eventually fail injectivity, creating a contradiction.

Key Contributions and Results
The primary result is Theorem 1.1, which establishes a strict dichotomy for any irreducible component $\Omega$ of the locus $V^k_n$:

• Alternative A: Every map $F \in \Omega$ is invertible (the Jacobian Conjecture holds for $\Omega$).
• Alternative B: The general map $F \in \Omega$ is non-invertible (the Jacobian Conjecture fails for $\Omega$).

The paper proves that there is no "middle ground" where a component contains both invertible and non-invertible maps in a way that prevents a generic classification. Specifically, the set of invertible maps within an irreducible component is either the entire component or a proper subvariety of lower dimension.

Furthermore, the paper connects this dichotomy to a probabilistic criterion based on the Ax-Grothendieck theorem (which states that injective polynomial maps are bijective). The author suggests that for a random choice of $(F, c) \in \Omega \times \mathbb{C}^n$, the solution to $F(z) = c$ exists and is unique if and only if the first alternative of the dichotomy holds.

Significance and Claims
The paper claims to provide an "organizing principle" that reframes the history of attempts to prove the Jacobian Conjecture. By establishing this dichotomy, the author suggests that the failure of previous general proofs may be due to the existence of components where the conjecture fails generically.

The significance lies in the potential for a computational approach to finding counterexamples. The author posits that if one applies the probabilistic criterion to all components of all $V^k_n$, one might locate a component $\Omega$ where the second alternative holds. In such a case, counterexamples would be "nearly impossible to miss" because they would constitute the generic behavior of that component. The paper does not claim to have found a counterexample or proven the conjecture true; rather, it provides a structural framework that narrows the search space for potential failures of the conjecture to specific irreducible components.

The work also references related results in real algebraic geometry regarding the invertibility of polynomial maps $G: \mathbb{R}^n \to \mathbb{R}^n$, utilizing semialgebraic sets and the Tarski-Seidenberg theorem to define a "kernel" set that vanishes if and only if the map is invertible.