Supertropical quadratic forms II

Tropical trigonometry and applications

Zur Izhakian, Manfred Knebusch, Louis Rowen

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

This paper is a sequel of [Z. Izhakian, M. Knebusch and L. Rowen, Supertropical semirings and supervaluations, J. Pure Appl. Algebra220(1) (2016) 61–93], where we introduced quadratic forms on a module VV over a supertropical semiring RR and analyzed the set of bilinear companions of a quadratic form q:V→Rq:V→R in case the module VV is free, with fairly complete results if RR is a supersemifield. Given such a companion bb, we now classify the pairs of vectors in VV in terms of (q,b).(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)CS(x,y) of a pair of anisotropic vectors x,yx,y in VV. We apply this to study the supertropicalizations (cf. [Z. Izhakian, M. Knebusch and L. Rowen, Supertropical semirings and supervaluations, J. Pure Appl. Algebra220(1) (2016) 61–93]) of a quadratic form on a free module XX over a field in the simplest cases of interest where rk(X)=2rk(X)=2.In the last part of the paper, we introduce a suitable equivalence relation on V∖{0}V∖{0}, whose classes we call rays. (It is coarser than usual projective equivalence.) For anisotropic x,y∈Vx,y∈V the CS-ratio CS(x,y)CS(x,y) depends only on the rays of xx and yy. We develop essential basics for a kind of convex geometry on the ray-space of VV, 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
Early online date10 Sep 2018
DOIs
Publication statusPublished - 30 Sep 2018

Fingerprint

Trigonometry
Semiring
Quadratic form
Half line
Module
Convex Geometry
Equivalence relation
Sort
Classify
Equivalence

Keywords

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

Cite this

Supertropical quadratic forms II : Tropical trigonometry and applications. / Izhakian, Zur; Knebusch, Manfred; Rowen, Louis.

In: International Journal of Algebra and Computation, Vol. 28, No. 8, 30.09.2018, p. 1633-1676.

Research output: Contribution to journalArticle

@article{fc5a2f5f80cc435aa25c4d34f00f2a3d,
title = "Supertropical quadratic forms II: Tropical trigonometry and applications",
abstract = "This paper is a sequel of [Z. Izhakian, M. Knebusch and L. Rowen, Supertropical semirings and supervaluations, J. Pure Appl. Algebra220(1) (2016) 61–93], where we introduced quadratic forms on a module VV over a supertropical semiring RR and analyzed the set of bilinear companions of a quadratic form q:V→Rq:V→R in case the module VV is free, with fairly complete results if RR is a supersemifield. Given such a companion bb, we now classify the pairs of vectors in VV in terms of (q,b).(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)CS(x,y) of a pair of anisotropic vectors x,yx,y in VV. We apply this to study the supertropicalizations (cf. [Z. Izhakian, M. Knebusch and L. Rowen, Supertropical semirings and supervaluations, J. Pure Appl. Algebra220(1) (2016) 61–93]) of a quadratic form on a free module XX over a field in the simplest cases of interest where rk(X)=2rk(X)=2.In the last part of the paper, we introduce a suitable equivalence relation on V∖{0}V∖{0}, whose classes we call rays. (It is coarser than usual projective equivalence.) For anisotropic x,y∈Vx,y∈V the CS-ratio CS(x,y)CS(x,y) depends only on the rays of xx and yy. We develop essential basics for a kind of convex geometry on the ray-space of VV, where the CS-ratios play a major role.",
keywords = "supertropicalization, Tropical algebra, supertropical modules, bilinear forms, quadratic forms, quadratic pairs, CS-ratio",
author = "Zur Izhakian and Manfred Knebusch and Louis Rowen",
note = "https://www.worldscientific.com/page/authors/author-rights#Preprint",
year = "2018",
month = "9",
day = "30",
doi = "10.1142/S021819671840012X",
language = "English",
volume = "28",
pages = "1633--1676",
journal = "International Journal of Algebra and Computation",
issn = "0218-1967",
publisher = "World Scientific Publishing Co. Pte Ltd",
number = "8",

}

TY - JOUR

T1 - Supertropical quadratic forms II

T2 - Tropical trigonometry and applications

AU - Izhakian, Zur

AU - Knebusch, Manfred

AU - Rowen, Louis

N1 - https://www.worldscientific.com/page/authors/author-rights#Preprint

PY - 2018/9/30

Y1 - 2018/9/30

N2 - This paper is a sequel of [Z. Izhakian, M. Knebusch and L. Rowen, Supertropical semirings and supervaluations, J. Pure Appl. Algebra220(1) (2016) 61–93], where we introduced quadratic forms on a module VV over a supertropical semiring RR and analyzed the set of bilinear companions of a quadratic form q:V→Rq:V→R in case the module VV is free, with fairly complete results if RR is a supersemifield. Given such a companion bb, we now classify the pairs of vectors in VV in terms of (q,b).(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)CS(x,y) of a pair of anisotropic vectors x,yx,y in VV. We apply this to study the supertropicalizations (cf. [Z. Izhakian, M. Knebusch and L. Rowen, Supertropical semirings and supervaluations, J. Pure Appl. Algebra220(1) (2016) 61–93]) of a quadratic form on a free module XX over a field in the simplest cases of interest where rk(X)=2rk(X)=2.In the last part of the paper, we introduce a suitable equivalence relation on V∖{0}V∖{0}, whose classes we call rays. (It is coarser than usual projective equivalence.) For anisotropic x,y∈Vx,y∈V the CS-ratio CS(x,y)CS(x,y) depends only on the rays of xx and yy. We develop essential basics for a kind of convex geometry on the ray-space of VV, where the CS-ratios play a major role.

AB - This paper is a sequel of [Z. Izhakian, M. Knebusch and L. Rowen, Supertropical semirings and supervaluations, J. Pure Appl. Algebra220(1) (2016) 61–93], where we introduced quadratic forms on a module VV over a supertropical semiring RR and analyzed the set of bilinear companions of a quadratic form q:V→Rq:V→R in case the module VV is free, with fairly complete results if RR is a supersemifield. Given such a companion bb, we now classify the pairs of vectors in VV in terms of (q,b).(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)CS(x,y) of a pair of anisotropic vectors x,yx,y in VV. We apply this to study the supertropicalizations (cf. [Z. Izhakian, M. Knebusch and L. Rowen, Supertropical semirings and supervaluations, J. Pure Appl. Algebra220(1) (2016) 61–93]) of a quadratic form on a free module XX over a field in the simplest cases of interest where rk(X)=2rk(X)=2.In the last part of the paper, we introduce a suitable equivalence relation on V∖{0}V∖{0}, whose classes we call rays. (It is coarser than usual projective equivalence.) For anisotropic x,y∈Vx,y∈V the CS-ratio CS(x,y)CS(x,y) depends only on the rays of xx and yy. We develop essential basics for a kind of convex geometry on the ray-space of VV, where the CS-ratios play a major role.

KW - supertropicalization

KW - Tropical algebra

KW - supertropical modules

KW - bilinear forms

KW - quadratic forms

KW - quadratic pairs

KW - CS-ratio

U2 - 10.1142/S021819671840012X

DO - 10.1142/S021819671840012X

M3 - Article

VL - 28

SP - 1633

EP - 1676

JO - International Journal of Algebra and Computation

JF - International Journal of Algebra and Computation

SN - 0218-1967

IS - 8

ER -