David Lippel : Positive elimination in valued fields.

Posted by , part of the Logic Seminar.

At

April 14, 2008, 4 p.m.

In

SEO 427

Abstract

A "positive elimination theorem" is a statement that certain positive existential formulas are equivalent to positive quantifier-free formulas. Here is a classical example. Let X be a Zariski-closed subset of a complex projective space. Concretely, X is the solution set of a system of homogeneous polynomial equations; thus, X has a positive quantifier-free definition in the language of rings. Let Y be a coordinate projection of X, so Y has a positive existential definition. Classical elimination theory says that Y is Zariski-closed, i.e. Y is actually defined by a positive quantifier-free formula. Prestel has proved some positive elimination results for valued fields, working in a one-sorted language. I will discuss some generalizations to two-sorted languages; these can be used to re-prove some basic facts in tropical geometry. This is joint work with Matthias Aschenbrenner and Sergei Starchenko.

tea at 4, talk shortly thereafter.