### Abstract

Original language | English |
---|---|

Pages (from-to) | 1633-1676 |

Number of pages | 44 |

Journal | International Journal of Algebra and Computation |

Volume | 28 |

Issue number | 8 |

Early online date | 10 Sep 2018 |

DOIs | |

Publication status | Published - 30 Sep 2018 |

### Fingerprint

### Keywords

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

### Cite this

*International Journal of Algebra and Computation*,

*28*(8), 1633-1676. https://doi.org/10.1142/S021819671840012X

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

Research output: Contribution to journal › Article

*International Journal of Algebra and Computation*, vol. 28, no. 8, pp. 1633-1676. https://doi.org/10.1142/S021819671840012X

}

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 -