Algebraic Geometry Seminar

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Tropical Varieties and Linear Algebra

Speaker: Louis Rowen, Bar-Ilan University

Location: Warren Weaver Hall 201

Date: Tuesday, February 17, 2015, 3:30 p.m.

Synopsis:

Tropicalization involves passing to an ordered group M, usually taken to be (R; +) or (Q; +) and viewed as a semifield. Although there is a rich theory arising from this viewpoint, idempotent semirings possess a restricted algebraic structure theory, and also do not reflect important valuation-theoretic properties, thereby forcing researchers to rely often on combinatoric techniques. Over the last few years, often jointly with Izhakian and Knebusch, we have studied an alternative structure, more compatible with valuation theory, that permits fuller use of algebraic structure in understanding the underlying tropical geometry. The idempotent max-plus algebra A of an ordered monoid M is replaced by R := L x M, where L is a given indexing semiring (not necessarily with 0). In this case we say that R is layered by L. When L is trivial, i.e. L = {1}, R is the usual bipotent max-plus algebra. When L = {1,∞} we recover the "standard" supertropical structure with its "ghost" layer. When L = N we can describe multiple roots of polynomials via a "layering function" s : R → L. Although self-contained, this talk continues the talk given in the spring of 2014 at Courant by Izhakian. We focus on how classical algebraic tools from algebraic geometry can be utilized in the supertropical theory. This includes resultants, discriminants, smoothness, etc. We also describe supertropical linear algebra. Recently, two related structures have been developed, one by Sheiner related to the "exploded" algebra and the other by Perri involving homomorphisms of ordered groups.