Quasilinear convexity and quasilinear stars in the ray space of a supertropical quadratic form

TL;DR

This paper develops a convex geometric framework for analyzing quasilinear submodules in supertropical quadratic forms using rays, convexity, and supertropical trigonometry, revealing rich combinatorial structures.

Contribution

It introduces a convex geometric approach based on rays and supertropical trigonometry to study quasilinear submodules of supertropical quadratic forms, with new criteria and structures.

Findings

Established a criterion for quasilinearity via Cauchy-Schwartz ratio.

Developed a convex geometry on Ray(V) supported by supertropical trigonometry.

Discovered rich combinatorial structures in minimal paths of quasilinear rays.

Abstract

Relying on rays, we search for submodules of a module V over a supertropical semiring on which a given anisotropic quadratic form is quasilinear. Rays are classes of a certain equivalence relation on V, that carry a notion of convexity, which is consistent with quasilinearity. A criterion for quasilinearity is specified by a Cauchy-Schwartz ratio which paves the way to a convex geometry on Ray(V), supported by a "supertropical trigonometry". Employing a (partial) quasiordering on Ray(V), this approach allows for producing convex quasilinear sets of rays, as well as paths, containing a given quasilinear set in a systematic way. Minimal paths are endowed with a surprisingly rich combinatorial structure, delivered to the graph determined by pairs of quasilinear rays -- apparently a fundamental object in the theory of supertropical quadratic forms.