Supertropical Quadratic Forms II

This article is a sequel of [4], where we introduced quadratic forms on a module~ $V$ over a supertropical semiring $R$ and analysed the set of bilinear companions of a quadratic form $q: V \to R$ in case that 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 that a sort of Cauchy-Schwarz inequality holds or fails. We apply this to study the supertropicalizations (cf. [4]) 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 start exploiting the fact that the free module $V$ as above has a unique base up to permutations and multiplication by units of $R$, and moreover~$V$ carries a so called minimal (partial) ordering. Under mild restriction on~$R$ we determine all $q$-minimal vectors in $V$, i.e., the vectors $x \in V$ for which $q(x') < q(x)$ whenever $x' < x.$