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Home » DDC - Introductory Seminar: Analogues of Hilbert’s Tenth Problems for rings of analytic functions and some open questions in Number Theory 2

Seminar

DDC - Introductory Seminar: Analogues of Hilbert’s Tenth Problems for rings of analytic functions and some open questions in Number Theory 2 November 20, 2020 (08:00 AM PST - 09:00 AM PST)
Parent Program:
Location: MSRI: Online/Virtual
Speaker(s) Thanases Pheidas (University of Crete)
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Video
Abstract/Media

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

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