Basic operations on supertropical quadratic forms
Summary: 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.
scientific article; zbMATH DE number 6836591
scientific article