Basic Operations on Supertropical Quadratic Forms

In the case that a module V over a (commutative) supertropical semiring R is free, the R-module Quad(V) of all quadratic forms on V is almost never a free module. Nevertheless, Quad(V) has two free submodules, the module QL(V) of quasilinear forms with base D₀ and the module Rig(V) of rigid forms with base H₀, such that Quad(V) = QL(V) + Rig(V) and QL(V) ∩ Rig(V) = {0}.In this paper we study endomorphisms of Quad(V) for which each submodule Rq with q ∈ D₀ ∪ H₀ is invariant; these basic endomorphisms are determined by coefficients in R and do not depend on the base of V. We aim for a description of all basic endomorphisms of Quad(V), or more generally of its submodules spanned by subsets of D₀ ∪ H₀. But, due to complexity issues, this naive goal is highly non-trivial for an arbitrary supertropical semiring R. Our main stress is therefore on results valid under only mild conditions on R, while a complete solution is provided for the case that R is a tangible super-semifield.