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Equations over Polyhedral Semirings

Madhusudan Manjunath

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.

Equations over Polyhedral Semirings

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.

Paper Structure

This paper contains 24 sections, 36 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Local Solutions
  3. Local-Global Principle
  4. Multiplicity and Discriminants
  5. Motivation and Related Work
  6. Polyhedral Semirings
  7. Polyhedralisation
  8. The Local Case
  9. Local Compatible Systems
  10. Vertex-Cone Collections
  11. Operations on Vertex-Cone Collections
  12. Polynomials and their Roots
  13. Proof of Theorem \ref{['genminwey_theo']}
  14. Completions, Associated Polyhedra and Shrinking
  15. Proof of Theorem \ref{['minweychar_theointro']}
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $\phi_{\rm gen}$ be a generic polyhedral polynomial. Fix a vertex $v$ of $M$. A polyhedron $P_0$ with vertex set $V$ is a $v$-local Minkowski-Weyl minimal solution to $\phi_{\rm gen}$ if and only if there is a local compatible system $(\mathcal{C},\mathcal{I})$ at $v$ where $\mathcal{C}=\{C_k\}_

Theorems & Definitions (83)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Remark 2.7
  • Proposition 3.1
  • ...and 73 more