The Algebraic Boundary of the Sonc-Cone
Jens Forsgård, Timo de Wolff
Abstract
In this article, we explore the connections between nonnegativity, the theory of $A$-discriminants, and tropical geometry. For an integral support set $A \subset \mathbb{Z}^n$, we cover the boundary of the sonc-cone by semialgebraic sets that are parametrized by families of tropical hypersurfaces. As an application, we give sufficient conditions for equality between the sonc-cone and the sparse nonnegativity cone for generic support sets, and we describe a semialgebraic stratification of the boundary of the sonc-cone in the univariate case.
Topics & Concepts
Cone (formal languages)MathematicsBoundary (topology)Algebraic numberCover (algebra)Stratification (seeds)Algebraic geometryPure mathematicsDual cone and polar coneGeometryCombinatoricsAlgebra over a fieldMathematical analysisAlgorithmRegular polygonSeed dormancyMechanical engineeringEngineeringBotanyDormancyBiologyGerminationPolynomial and algebraic computationAdvanced Differential Equations and Dynamical SystemsAdvanced Numerical Analysis Techniques