Supertropical quadratic forms I

Zur Izhakian, Manfred Knebusch, Louis Rowen*

*Corresponding author for this work

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

We initiate the theory of a quadratic form q over a semiring R, with a view to study tropical linear algebra. As customary, one can writeq(x+y)=q(x)+q(y)+b(x,y), where b is a companion bilinear form. In contrast to the classical theory of quadratic forms over a field, the companion bilinear form need not be uniquely defined. Nevertheless, q can always be written as a sum of quadratic forms q=qQL+ρ, where qQL is quasilinear in the sense that qQL(x+y)=qQL(x)+qQL(y), and ρ is rigid in the sense that it has a unique companion. In case that R is supertropical, we obtain an explicit classification of these decompositions q=qQL+ρ and of all companions b of q, and see how this relates to the tropicalization procedure.

Original languageEnglish
Pages (from-to)61-93
Number of pages33
JournalJournal of Pure and Applied Algebra
Volume220
Issue number1
Early online date12 Jun 2015
DOIs
Publication statusPublished - 1 Jan 2016

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Quadratic form
Bilinear form
Semiring
Linear algebra
Decompose

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

Supertropical quadratic forms I. / Izhakian, Zur; Knebusch, Manfred; Rowen, Louis.

In: Journal of Pure and Applied Algebra, Vol. 220, No. 1, 01.01.2016, p. 61-93.

Research output: Contribution to journalArticle

Izhakian, Zur ; Knebusch, Manfred ; Rowen, Louis. / Supertropical quadratic forms I. In: Journal of Pure and Applied Algebra. 2016 ; Vol. 220, No. 1. pp. 61-93.
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