Basic Operations on Supertropical Quadratic Forms

Abstract

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 ${\mathfrak{D}}_0$ and the module Rig($V$) of rigid forms with base ${\mathfrak{H}}_0$, such that Quad($V$)=QL($V$)+Rig($V$) and QL($V$)$\cap$ Rig($V$) =$\{0\}$. In this paper we study endomorphisms of Quad($V$) for which each submodule $Rq$ with $q\in {\mathfrak{D}}_0\cup {\mathfrak{H}}_0$ 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 ${\mathfrak{D}}_0\cup {\mathfrak{H}}_0$. But, due to complexity issues, this naive goal is highly nontrivial 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 supersemifield.