A Tate algebra version of the Jacobian Conjecture

TL;DR

This paper explores a Tate algebra variant of the Jacobian conjecture, establishing conditions under which it is equivalent to the classical conjecture and analyzing its validity across different characteristics.

Contribution

It introduces the Tate-Jacobian conjecture, proving its equivalence to the Jacobian conjecture under certain topological and algebraic conditions, and relates it to polynomial invertibility over p-adic fields.

Findings

Tate-Jacobian conjecture is equivalent to the Jacobian conjecture when R/I is in a Q-algebra.

The conjecture fails in positive characteristic cases.

The Jacobian conjecture over C is linked to invertibility over almost all p-adic fields.

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 $\mathbb{C}_p$.