arXiv subjects

Trevor Gunn

Publications and source records attributed to Trevor Gunn.

4 recordsLinked to original sources

arXiv2023

Factoring multivariate polynomials over hyperfields and the multivariable Descartes' problem

We develop several notions of multiplicity for linear factors of multivariable polynomials over different arithmetics (hyperfields). The key example is multiplicities over the hyperfield of signs, which encapsulates the arithmetic of $\mathbf{R}/\mathbf{R}_{>0}$. These multiplicities give us various upper and lower bounds on the number of linear factors with a given sign pattern in terms of the signs of the coefficients of the factored polynomial. Using resultants, we can transform a square system of polynomials into a single polynomial whose multiplicities give us bounds on the number of positive solutions to the system. In particular, we are able to re-derive the lower bound of Itenberg and Roy on any potential upper bound for the number of solutions to a system of equations with a given sign pattern. In addition, our techniques also explain a particular counterexample of Li and Wang to Itenberg and Roy's proposed upper bound.

math.AG↗

arXiv2023

Tropical Extensions and Baker-Lorscheid Multiplicities for Idylls

In a recent paper, Matthew Baker and Oliver Lorscheid showed that Descartes's Rule of Signs and Newton's Polygon Rule can both be interpreted as multiplicities of polynomials over hyperfields. Hyperfields are a generalization of fields which encode things like the arithmetic of signs or of absolute values. By looking at multiplicities of polynomials over such algebras, Baker and Lorscheid showed that you can recover the rules of Descartes and Newton. In this paper, we define tropical extensions for idylls. Such extensions have appeared for semirings with negation symmetries in the work of Akian-Gaubert-Guterman and for hypergroups and hyperfields in the work of Bowler-Su. Examples of tropical extensions are extending the tropical hyperfield to higher ranks, or extending the hyperfield of signs to the tropical real hyperfield by including a valuation. The results of this paper concern the interaction of multiplicities and tropical extensions. First, there is a lifting theorem from initial forms to the entire polynomial from which we will show that multiplicities for a polynomial are equal to the corresponding multiplicity for some initial form. Second, we show that tropical extensions preserve the property that the sum of all multiplicities is bounded by the degree. Consequentially, we have this degree bound for every stringent hyperfield. This gives a partial answer to a question posed by Baker and Lorscheid about which hyperfields have this property.

math.RA↗

arXiv2022

Construction of Fully Faithful Tropicalizations for Curves in Ambient Dimension 3

In tropical geometry, one studies algebraic curves using combinatorial techniques via the tropicalization procedure. The tropicalization depends on a map to an algebraic torus and the combinatorial methods are most useful when the tropicalization has nice properties. We construct, for any Mumford curve $X$, a map to a three-dimensional torus, such that the tropicalization is isometric to a subgraph of the Berkovich space $X^{\rm an}$, called the extended skeleton. In this case, we say the tropicalization is "fully faithful." Additionally, given a map $X$ to a toric variety $Y$, which induces a fully faithful tropicalization, we show that we can extend the map to $X \to Y \times (\mathbf{P}^1)^n$ such that the new tropicalization is smooth and fully faithful.

math.AG↗

arXiv2021

A Newton Polygon Rule for Formally-Real Valued Fields and Multiplicities over the Signed Tropical Hyperfield

By defining multiplicities for zeros of polynomials over hyperfields, Baker and Lorscheid were able to provide a unifying perspective on Descartes's rule and the Newton polygon rule for polynomials over a formally-real and valued field respectively. In this paper, we apply their multiplicity formula to the hyperfield associated with formally-real, valued fields to prove a Newton polygon rule which combines Descartes's rule of signs with the classical Newton polygon rule.

math.RA↗