Existential first-order definitions of valuations in function fields
Date:
Abstract: Starting from an arbitrary field K, one may form the rational function field K(T): the field of fractions of polynomials in one variable over K. K(T) naturally carries infinitely many K-trivial discrete valuations. One example of such a valuation is the so-called `degree valuation’, whose valuation ring consists of those rational functions f/g given by polynomials f and g satisfying deg(f) ≤ deg(g).
In 1978 Denef showed that, if K is the field of rational or real numbers, then the valuation ring of the degree valuation has an existential first-order definition in K(T) in the language of rings. His proof crucially relied on the ordered structure on the base field. In the decades thereafter, a myriad of different types of base fields K have been considered for which existential definability of the degree valuation in K(T) could be proven. Perhaps somewhat curiously, the used constructions and techniques all depend heavily on arithmetic properties of the base field K. To this day, no unified theory explaining all known examples seems to have been formulated.
In this talk, I will give a proof of a case of Denef’s original result, discuss how parts of the method can be generalized, and where obstacles lie for our understanding of a general solution. Insofar time allows, this talk may discuss joint work with Karim Johannes Becher and Philip Dittmann.
This talk was part of the a workshop of the Trimester Program: “Definability, decidability, and computability” at the Hausdorff Research Institute for Mathematics in Bonn. The recording of this talk may be found here.