Thursday, October 2
Milner 317, 1:00 PM
Title: Hilbert's Tenth Problem for function fields of surfaces over C
Abstract: Hilbert's Tenth Problem in its original form is the following: Is there a uniform algorithm that determines, given a multivariate polynomial equation with integer coefficients, whether the equation has a solution over the integers?
Davis-Putnam-Robinson-Matiyasevich showed that there is no such algorithm, i.e. that Hilbert's Tenth Problem is undecidable. Since then the analogue of this question has been studied for various rings. I will discuss the result by Kim and Roush that Hilbert's Tenth Problem for C(t_1,t_2) is undecidable, and show how their result can be generalized to finite extensions of C(t_1,t_2), i.e. to function fields of surfaces over the complex numbers.