Supertropical quadratic forms II: Tropical trigonometry and applications

This paper is a sequel of [Z. Izhakian, M. Knebusch and L. Rowen, Supertropical semirings and supervaluations, J. Pure Appl. Algebra 220(1) (2016) 61-93], where we introduced quadratic forms on a module V over a supertropical semiring R and analyzed the set of bilinear companions of a quadratic form q: V → R in case the module V is free, with fairly complete results if R is a supersemifield. Given such a companion b, we now classify the pairs of vectors in V in terms of (q,b). This amounts to a kind of tropical trigonometry with a sharp distinction between the cases for which a sort of Cauchy-Schwarz (CS) inequality holds or fails. This distinction is governed by the so-called CS-ratio CS(x,y) of a pair of anisotropic vectors x,y in V. We apply this to study the supertropicalizations (cf. [Z. Izhakian, M. Knebusch and L. Rowen, Supertropical semirings and supervaluations, J. Pure Appl. Algebra 220(1) (2016) 61-93]) of a quadratic form on a free module X over a field in the simplest cases of interest where rk(X) = 2. In the last part of the paper, we introduce a suitable equivalence relation on V \{0}, whose classes we call rays. (It is coarser than usual projective equivalence.) For anisotropic x,y ϵ V the CS-ratio CS(x,y) depends only on the rays of x and y. We develop essential basics for a kind of convex geometry on the ray-space of V, where the CS-ratios play a major role.