Supertropical Quadratic Forms II

TL;DR

This paper extends the theory of quadratic forms over supertropical semirings, classifies vector pairs via tropical trigonometry, and explores minimal vectors, with applications to supertropicalizations of forms over fields.

Contribution

It introduces a classification of vector pairs using quadratic forms and bilinear companions, and analyzes minimal vectors in supertropical modules, advancing the understanding of supertropical quadratic forms.

Findings

Classification of vector pairs in terms of quadratic forms and companions.

Identification of conditions for Cauchy-Schwarz inequality to hold or fail.

Determination of all q-minimal vectors in supertropical modules.

Abstract

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 $x' < x.$