dc.contributor.author Knebusch, Manfred dc.contributor.author Rowen, Louis dc.contributor.author Izhakian, Zur dc.date.accessioned 2016-09-22T10:46:39Z dc.date.available 2016-09-22T10:46:39Z dc.date.issued 2013 dc.identifier.uri http://publications.mfo.de/handle/mfo/200 dc.description OWLF 2013 en_US dc.description.abstract We initiate the theory of a quadratic form q over a semiring $R$. As customary, one can write $q(x+y)=q(x)+q(y)+b(x,y)$, where b is a companion bilinear form. But in contrast to the ring-theoretic case, the companion bilinear form need not be uniquely defined. Nevertheless, q can always be written as a sum of quadratic forms $q=\kappa+\rho$, where $\kappa$ is quasilinear in the sense that $\kappa(x+y)=\kappa(x)+\kappa(y)$, and $\rho$ is rigid in the sense that it has a unique companion. In case that $R$ is a supersemifield (cf. Definition 4.1 below) and $q$ is defined on a free $R$-module, we obtain an explicit classification of these decompositions $q=\kappa+\rho$ and of all companions $b$ of $q$. As an application to tropical geometry, given a quadratic form $q:V \to R$ on a free module $V$ over a commutative ring $R$ and a supervaluation $\rho$: $R \to U$ with values in a supertropical semiring [5], we define - after choosing a base $\mathcal{L}=(v_i|i \in I)$ of $V$- a quadratic form $q^\varphi:U^{(I)} \to U$ on the free module $U^{(I)}$ over the semiring $U$. The analysis of quadratic forms over a supertropical semiring enables one to measure the “position” of $q$ with respect to $\mathcal{L}$ via ${\varphi}$. en_US dc.language.iso en en_US dc.publisher Mathematisches Forschungsinstitut Oberwolfach en_US dc.relation.ispartofseries Oberwolfach Preprints;2013,27 dc.subject Tropical Algebra en_US dc.subject Supertropical Modules en_US dc.subject Bilinear Forms en_US dc.subject Quadratic Forms en_US dc.subject Quadratic Pairs en_US dc.subject Supertropicalization en_US dc.title Supertropical Quadratic Forms I en_US dc.type Preprint en_US dc.rights.license Dieses Dokument darf im Rahmen von § 53 UrhG zum eigenen Gebrauch kostenfrei heruntergeladen, gelesen, gespeichert und ausgedruckt, aber nicht im Internet bereitgestellt oder an Außenstehende weitergegeben werden. de dc.rights.license This document may be downloaded, read, stored and printed for your own use within the limits of § 53 UrhG but it may not be distributed via the internet or passed on to external parties. en dc.identifier.doi 10.14760/OWP-2013-27 local.scientificprogram OWLF 2013 local.series.id OWP-2013-27 local.subject.msc 11 local.subject.msc 15 local.subject.msc 16 local.subject.msc 14 local.subject.msc 13 dc.identifier.urn urn:nbn:de:101:1-2014013014022 dc.identifier.ppn 1653203668
﻿