### Divisors and multiplicities under tropical and signed shadows

- Series
- Dissertation Defense
- Time
- Tuesday, June 20, 2023 - 09:30 for 1.5 hours (actually 80 minutes)
- Location
- Skiles 006 / Zoom
- Speaker
- Trevor Gunn – Georgia Tech – tgunn@gatech.edu

Zoom link (Meeting ID: 941 5991 7033, Passcode: 328576)

I will present two projects related to tropical divisors and multiplicities. First, my work with Philipp Jell on fully-faithful tropicalizations in 3-dimensions. Second, my work with Andreas Gross on algebraic and combinatorial multiplicities for multivariate polynomials over the tropical and sign hyperfields.

The first part is about using piecewise linear functions to describe tropical curves in 3 dimensions and how the changes in those slopes (a divisor) lift to non-Archimedean curves. These divisors give an embedding of a curve in a 3-dimensional toric variety whose tropicalization is isometric to the so-called extended skeleton of the curve.

In part two, I describe how Baker and Lorscheid's theory of multiplicities over hyperfields can be extended to multivariate polynomials. One key result is a new proof/view of the work of Itenburg and Roy who used patchworking to construct some lower bounds on the number of positive roots of a system of polynomials given a particular sign arrangement. Another result is a collection of upper bounds for the same problem.

**Committee:**

- Matt Baker (Advisor)
- Josephine Yu
- Oliver Lorscheid
- Anton Leykin
- Greg Blekherman