Basic operations on supertropical quadratic forms
type: Item
instance of: scholarly article
title: Basic operations on supertropical quadratic forms
zbMATH Open document ID: 1388.16045
author: Zur Izhakian
author: Manfred Knebusch
published in: Documenta Mathematica
publication date: 2018-02-09T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 16Y60
Mathematics Subject Classification ID: 14T05
Mathematics Subject Classification ID: 20M18
Mathematics Subject Classification ID: 15A63
zbMATH DE Number: 6836591
zbMATH Keywords: tropical algebra
zbMATH Keywords: supertropical modules
zbMATH Keywords: bilinear forms
zbMATH Keywords: quadratic forms
zbMATH Keywords: quadratic pairs
zbMATH Keywords: minimal ordering
zbMATH Keywords: unique base property
MaRDI profile type: Publication
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DOI: 10.25537/DM.2017V22.1661-1707
description: scientific article; zbMATH DE number 6836591