A Tate algebra version of the Jacobian conjecture

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 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 C is equivalent to the following statement: for all but finitely many primes p , the inverse of a polynomial map over C p whose Jacobian determinant is an element of C p × lies in the Tate algebra over C p .