Minimal orderings and quadratic forms on a free module over a supertropical semiring

This paper is a sequel to [6], in which we introduced quadratic forms on a module over a supertropical semiring R and analyzed the set of bilinear companions of a single quadratic form V→R in case the module V is free. Any (semi)module over a semiring gives rise to what we call its minimal ordering, which is a partial order iff the semiring is “upper bound.” Any polynomial map q (or quadratic form) then induces a pre-order, which can be studied in terms of “q-minimal elements,” which are elements a which cannot be written in the form b+c where b<a but q(b)=q(a). We determine the q-minimal elements by examining their support. But the class of all polynomial maps (in up to rank(V) variables) is itself a module over R, so the basic properties of the minimal ordering are applied to this R-module, or its submodule Quad(V) consisting of quadratic forms on V. This is a significant initial step in the classification of quadratic forms over semirings arising in tropical mathematics. Quad(V) is the sum of two disjoint submodules QL(V) and Rig(V), consisting of the quasilinear and the rigid quadratic forms on V respectively (cf. [6]). Both QL(V) and Rig(V) are free with explicitly known bases, but Quad(V) itself is almost never free.