Seminar in Real & Complex Geometry

Sunday, 04.08.2013, 16:15-17:30, Schreiber building, room 209

Tal Perri, Bar-Ilan University

Algebraic aspects in tropical mathematics


In our study, much like the theory of algebraic geometry, we have established a connection between tropical geometry and algebra. Namely, we establish a correspondence between tropical varieties and an algebraic structures named "kernels" in the semifield of fractions (viewed as a subset of the semifield of functions over the max-plus semifield). The kernels serve as the analogue to ideals while the semifield of fractions takes the place of the polynomial ring. We have found a way to represent the corner loci defined by a tropical polynomial by an equation in the semifield of fractions. The resulting equation gives rise to a congruence which is encapsulated in the structure of a kernel (like zero sets and ideals in algebraic geometry). Being an idempotent semifield, the semifield of fractions is also a "lattice ordered group" (a group whose elements are endowed with a lattice structure compatible with the group operation) and the kernels are normal convex l-subgroups there. The theory of lattice ordered group has been studied extensively by Birkhoff, Medvedev, Kopytov, Steinberg, Anderson, Feil and many other mathmaticials. It offers a whole new set of algebraic tools to study tropical varieties.