Title:
Divisors and multiplicities under tropical and signed shadows
Divisors and multiplicities under tropical and signed shadows
Author(s)
Gunn, Trevor
Advisor(s)
Baker, Matthew
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Abstract
This thesis addresses questions related to divisors and multiplicities as analyzed through tropicalization or signs. It begins with a introduction to the subject matter written for a non-specialist. The next chapter concerns fully-faithful tropicalization in low dimension. The last two chapters concern questions about Baker-Lorscheid multiplicities in one and several variables respectively.
With fully-faithful tropicalization, the goal was to construct a tropicalization map from a curve to a 3-dimensional toric variety. The constraints are that we need the map to be injective and we need the gcd of all the slopes to be 1, so that we get an isometry with respect to the lattice length metric. We also have some results about smooth, fully-faithful tropicalizations of a genus g curve in a toric variety of a dimension 2g + 2 (three more than the lower bound imposed by the maximal vertex degree).
For multiplicities, I present a broad generalization of the work of Baker and Lorscheid for univariate multiplicities over hyperfields. In Baker and Lorscheid's work, they show how Descartes's Rule of Signs and Newton's Polygon Rule may be obtained from factorizing polynomials in the arithmetics of signs and tropical numbers respectively. In Chapter 3, I introduce a broad generalization of their multiplicity operator to a class of arithmetics, which I call "whole-idylls." In particular, we have a way of extending multiplicity rules by extending the arithmetic by a valuation. An important corollary is that for so-called "stringent" hyperfields, we have a degree bound: the sum of multiplicities for a polynomial is bounded by its degree.
The last chapter contains my work with Andreas Gross on multivariate hyperfield multiplicities. We give particular attention to the hyperfield of signs and the so-far-unresolved Multivariate Descartes Question. We define several multiplicity operators for linear factors of polynomials and apply them to systems of equations. We recover the lower bound of Itenberg-Roy on any potential upper bound for roots with a given sign pattern.
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Date Issued
2023-07-24
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Dissertation