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. Algebra220(1) (2016) 61–93], where we introduced quadratic forms on a module VV over a supertropical semiring RR and analyzed the set of bilinear companions of a quadratic form q:V→Rq:V→R in case the module VV is free, with fairly complete results if RR is a supersemifield. Given such a companion bb, we now classify the pairs of vectors in VV in terms of (q,b).(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)CS(x,y) of a pair of anisotropic vectors x,yx,y in VV. We apply this to study the supertropicalizations (cf. [Z. Izhakian, M. Knebusch and L. Rowen, Supertropical semirings and supervaluations, J. Pure Appl. Algebra220(1) (2016) 61–93]) of a quadratic form on a free module XX over a field in the simplest cases of interest where rk(X)=2rk(X)=2.In the last part of the paper, we introduce a suitable equivalence relation on V∖{0}V∖{0}, whose classes we call rays. (It is coarser than usual projective equivalence.) For anisotropic x,y∈Vx,y∈V the CS-ratio CS(x,y)CS(x,y) depends only on the rays of xx and yy. We develop essential basics for a kind of convex geometry on the ray-space of VV, where the CS-ratios play a major role.