## 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 language | English |
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Pages (from-to) | 1633-1676 |

Number of pages | 44 |

Journal | International Journal of Algebra and Computation |

Volume | 28 |

Issue number | 8 |

DOIs | |

State | Published - 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