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dc.contributor.authorKnebusch, Manfred
dc.contributor.authorRowen, Louis
dc.contributor.authorIzhakian, Zur
dc.date.accessioned2016-09-22T10:46:39Z
dc.date.available2016-09-22T10:46:39Z
dc.date.issued2013
dc.identifier.urihttp://publications.mfo.de/handle/mfo/200
dc.descriptionOWLF 2013en_US
dc.description.abstractWe 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.isoenen_US
dc.publisherMathematisches Forschungsinstitut Oberwolfachen_US
dc.relation.ispartofseriesOberwolfach Preprints;2013,27
dc.subjectTropical Algebraen_US
dc.subjectSupertropical Modulesen_US
dc.subjectBilinear Formsen_US
dc.subjectQuadratic Formsen_US
dc.subjectQuadratic Pairsen_US
dc.subjectSupertropicalizationen_US
dc.titleSupertropical Quadratic Forms Ien_US
dc.typePreprinten_US
dc.rights.licenseDieses 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.licenseThis 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.doi10.14760/OWP-2013-27
local.scientificprogramOWLF 2013
local.series.idOWP-2013-27
local.subject.msc11
local.subject.msc15
local.subject.msc16
local.subject.msc14
local.subject.msc13
dc.identifier.urnurn:nbn:de:101:1-2014013014022
dc.identifier.ppn1653203668


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