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

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.