To participate in this seminar, please register here: https://www.msri.org/seminars/25206

Abstract:

Let A (D) be the ring of functions of an array, z,  of variables, as these range in an open superset of a power of the set D, which will be either the complex numbers or p-adic complex numbers, or the unit disc (open or closed)  of any of these. We ask:



Question: Is there an algorithm which determines whether or not any given polynomial equation, in an array x of variables, with coefficients in Z[z], has or does not have solutions in A(D)?  (Z is the ring of integers).



The answer is negative if D is the ring of p-adic complex numbers, for any prime number p, and any number of variables in z. It is open for z being one variable and D the ring of complex numbers or the unit disc. We will present ideas behind a negative answer to the question for z being two variables over the complex numbers, but in a language that allows evaluation at a point ("initial value poblems”).

We will show the relevance of an analogue of the question to some conjectures of S. Lang

Analogues Of Hilbert’s Tenth Problems For Rings Of Analytic Functions And Some Open Questions In Number Theory 2