Pastures, Polynomials, and Matroids

A pasture is, roughly speaking, a field in which addition is allowed to be both multivalued and partially undefined. Pastures are natural objects from the point of view of F_1 geometry and Lorscheid’s theory of ordered blueprints. I will describe a theorem about univariate polynomials over pastures which simultaneously generalizes Descartes’ Rule of Signs and the theory of Newton polygons. Conjecturally, there should be a similar picture for several polynomials in several variables generalizing tropical intersection theory. I will also describe a novel approach to the theory of matroid representations which revolves around a canonical universal pasture, called the “foundation”, that one can attach to any matroid. This is joint work with Oliver Lorscheid.

The discussion for Matt Baker’s talk is taking place not in zoom-chat, but at  https://tinyurl.com/2021-04-02-mb (and will be deleted after ~3-7 days).