Supertropical quadratic forms II: Tropical trigonometry and applications

Zur Izhakian, Manfred Knebusch, Louis Rowen

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Abstract

This paper is a sequel of [Z. Izhakian, M. Knebusch and L. Rowen, Supertropical semirings and supervaluations, J. Pure Appl. Algebra 220(1) (2016) 61-93], where we introduced quadratic forms on a module V over a supertropical semiring R and analyzed the set of bilinear companions of a quadratic form q: V → R in case the module V is free, with fairly complete results if R is a supersemifield. Given such a companion b, we now classify the pairs of vectors in V in terms of (q,b). This amounts to a kind of tropical trigonometry with a sharp distinction between the cases for which a sort of Cauchy-Schwarz (CS) inequality holds or fails. This distinction is governed by the so-called CS-ratio CS(x,y) of a pair of anisotropic vectors x,y in V. We apply this to study the supertropicalizations (cf. [Z. Izhakian, M. Knebusch and L. Rowen, Supertropical semirings and supervaluations, J. Pure Appl. Algebra 220(1) (2016) 61-93]) of a quadratic form on a free module X over a field in the simplest cases of interest where rk(X) = 2. In the last part of the paper, we introduce a suitable equivalence relation on V \{0}, whose classes we call rays. (It is coarser than usual projective equivalence.) For anisotropic x,y ϵ V the CS-ratio CS(x,y) depends only on the rays of x and y. We develop essential basics for a kind of convex geometry on the ray-space of V, where the CS-ratios play a major role.

Original languageEnglish
Pages (from-to)1633-1676
Number of pages44
JournalInternational Journal of Algebra and Computation
Volume28
Issue number8
DOIs
StatePublished - 1 Dec 2018

Bibliographical note

Publisher Copyright:
© 2018 World Scientific Publishing Company.

Keywords

  • CS-ratio
  • Tropical algebra
  • bilinear forms
  • quadratic forms
  • quadratic pairs
  • supertropical modules
  • supertropicalization

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