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# Jeffrey Giansiracusa : Equations of tropical varieties

Tropical geometry is a combinatorial shadow of algebraic geometry over a nonarchimedean field that encodes information about things like intersections and enumerative invariants. Usually one defines tropical varieties as certain polyhedral subsets of R^n satisfying a balancing condition. I'll show how these arise as the solution sets to certain systems of polynomial equations over the tropical semiring T = (R union -infinity, max, +) related to matroids. This yields a notion of tropical Hilbert polynomials, and in this framework there is a universal tropicalization that is closely related to the Berkovich analytification and the moduli space of valuations.

**Category**: Algebraic Geometry**Duration**: 04:47**Date**: April 30, 2015 at 2:55 PM-
**Tags:**seminar, Algebraic Geometry Seminar

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