Paving tropical ideals

Abstract: Tropical geometry is a powerful tool in algebraic geometry, which offers a multitude of combinatorial approaches to studying algebraic varieties. This talk will focus on the recent development of tropical commutative algebra by Diane Maclagan and Felipe Rincon. The central object of study is the “tropical Ideal,” which generalizes the structure of polynomial ideals over fields to be suitable for study in the setting of tropical geometry, that is, in polynomial semirings over semifields. All polynomial ideals over a field can be associated to a “realizable” tropical ideal, and it is a non-trivial fact that “non-realizable” tropical ideals exist. In this talk, I will demonstrate how the combinatorics of matroid theory allows us to easily generate a subclass of tropical ideals, called paving tropical ideals, which in turn allows us to prove that most zero-dimensional tropical ideals are not realizable.

algebraic geometrycombinatorics

Audience: researchers in the topic

Online Nottingham algebraic geometry seminar

Series comments: Online geometry seminar, typically held on Thursday. This seminar takes place online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team.

For recordings of past talks, copies of the speaker's slides, or to be added to the Team, please visit the seminar homepage at: kasprzyk.work/seminars/ag.html