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