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→RV→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+cb+c where b