A Tate algebra version of the Jacobian Conjecture

TL;DR

The paper develops a Tate algebra framework for a Jacobian-conjecture analogue and shows that the Tate-Jacobian conjecture $\mathrm{TJC}(R,I,n)$ is equivalent to the classical Jacobian conjecture $\mathrm{JC}(R/I,n)$ when the $I$-adic topology is Hausdorff, while failing in positive characteristic. It further connects JC over $\mathbb{C}$ to a p-adic criterion that must hold for all but finitely many primes $p$, and proves that JC over $\mathbb{C}$ is equivalent to the Unimodular conjecture for $\mathbb{Z}_p$ for almost all $p$, using a key lemma about inverses in Tate algebras and results of van den Essen and Lipton. This work links polynomial automorphisms over characteristic-zero fields to $p$-adic Tate-algebra methods, offering a new avenue to study JC via topological-algebraic techniques.

Abstract

This paper investigates a Tate algebra version of the Jacobian conjecture, referred to as the Tate-Jacobian conjecture, for commutative rings $R$ equipped with an $I$-adic topology. We show that if the $I$-adic topology on $R$ is Hausdorff and $R/I$ is a subring of a $\mathbb{Q}$-algebra, then the Tate-Jacobian conjecture is equivalent to the Jacobian conjecture. Conversely, if $R/I$ has positive characteristic, the Tate-Jacobian conjecture fails. Furthermore, we establish that the Jacobian conjecture for $\mathbb{C}$ is equivalent to the following statement: for all but finitely many primes $p$, the inverse of a polynomial map over $\mathbb{C}_p$ whose Jacobian determinant is an element of $\mathbb{C}_p^\times$ lies in the Tate algebra over $\mathbb{C}_p$.

Theorems & Definitions (5)

• proof
• proof
• proof : Proof of Theorem \ref{['thm: TJ conjecture for char zero']}
• proof
• proof : Proof of Theorem \ref{['main theorem 1']}