University Seminar- Computability, Complexity, and Algebraic Structure

Time: Wednesday, September 27, 11:15am – 12:15pm

Place: Monroe Hall, Room 250

Speaker: Dong Quan Nguyen, GWU

Title:  Hilbert’s Tenth Problem for the rational function field over an infinite field, and related definability problems

Abstract:  Hilbert's Tenth Problem asks for an algorithm, which for any given polynomial equation with integral coefficients and a finite number of variables, can decide whether the equation has a solution with all variables taking integer values. Due to the work of Davis, Putnam, Robinson, and Matiyasevich during 1950’s–70’s, Hilbert's Tenth Problem for the ring of integers has a negative answer. It is natural to consider an analogue of Hilbert's Tenth Problem for other rings or fields, among which there are two main open problems: the field of rationals, and the rational function field over the complex numbers. Regarding the rational function field C(t) with C being the complex numbers, even the full first-order theory of C(t) is not known to be decidable or not. In this talk, I will discuss my recent work which proves the undecidability of the full first-order theory of the rational function field F(t), where F is a nonprincipal ultraproduct of finite fields of unbounded characteristics. This follows from the definability of the polynomial ring F[t] in F(t). Even the case when F has a positive characteristic is new. I will mention the main tool used in the proof, which is a generalization of higher reciprocity law and Hilbert's reciprocity law for F(t) which I proved in another work, using model theory and theory of local symbols.