Tropical Extensions and Baker-Lorscheid Multiplicities for Idylls

TL;DR

This paper introduces tropical extensions for idylls, explores how multiplicities behave under these extensions, and shows that certain hyperfields maintain degree bounds for polynomial multiplicities, extending prior work on hyperfields and algebraic rules.

Contribution

It defines tropical extensions for idylls and proves that multiplicities are preserved and bounded by degree in these extensions, addressing a question by Baker and Lorscheid.

Findings

Multiplicities for polynomials are equal to those of initial forms.

Tropical extensions preserve the degree bound on the sum of multiplicities.

Degree bounds hold for every stringent hyperfield.

Abstract

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.