Izhakian, Zur and Knebusch, Manfred and Rowen, Louis (2018) Supertropical quadratic forms II: Tropical trigonometry and applications. INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION, 28 (8). pp. 1633-1676. ISSN 0218-1967, 1793-6500
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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 is an element of 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.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | ; Tropical algebra; supertropical modules; bilinear forms; quadratic forms; quadratic pairs; CS-ratio; supertropicalization |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics > Professoren und akademische Räte im Ruhestand > Prof. Dr. Manfred Knebusch |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 10 May 2022 06:52 |
| Last Modified: | 10 May 2022 06:52 |
| URI: | https://pred.uni-regensburg.de/id/eprint/13391 |
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