Valuation theory plays a major part in the interaction between number theory and logic. In this seminar, a variety of topics from valuation theory, and in particular connections to model theory, will be discussed. There will be a strong emphasis on results involving the definability of valuations.

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

Abstract:

In the context of extensions of Hilbert's tenth problem, one of the main open problems is the case of the ring of complex entire functions in one variable. Progress on this matter has been difficult due to the existence of the exponential function. In this talk I'll outline the proof of a negative solution for the ring of holomorphic exponential polynomials. The technique involves a mix of number theory, complex analysis, and functional transcendence. At the end, defining the valuation offers additional difficulties since the standard divisibility argument does not work in these rings. This is joint work with D. Chompitaki, N. Garcia-Fritz, T. Pheidas and X. Vidaux initiated at the 2019 AIM meeting on H10.