Equations over Polyhedral Semirings

TL;DR

This paper explores the solutions of polynomial equations over polyhedral semirings, providing characterizations, principles, and foundational concepts inspired by tropical geometry and physics.

Contribution

It introduces a framework for solving equations over polyhedral semirings, including local solutions, a local-global principle, and concepts of multiplicity and discriminants.

Findings

Characterization of local solutions based on coefficients

Establishment of a local-global solution principle

Development of foundational concepts like multiplicity and discriminants

Abstract

We study the theory of equations in one variable over polyhedral semirings. The article revolves around a notion of solution to a polynomial equation over a polyhedral semiring. Our main results are a characterisation of local solutions in terms of the coefficients, a local-global principle, and the basics of multiplicity and discriminants. Our primary sources of motivation are tropical geometry and the theory of exceptional points in non-Hermitian physics.