## Abstract

Relying on rays, we search for submodules of a module V over a supertropical semiring on which a given anisotropic quadratic form is quasilinear. Rays are classes of a certain equivalence relation on V, that carry a notion of convexity, which is consistent with quasilinearity. A criterion for quasilinearity is specified by a Cauchy-Schwartz ratio which paves the way to a convex geometry on (Formula presented.), supported by a ‘supertropical trigonometry’. Employing a (partial) quasiordering on (Formula presented.), this approach allows for producing convex quasilinear sets of rays, as well as paths, which contain a given quasilinear set in a systematic way. Minimal paths are endowed with a surprisingly rich combinatorial structure, delivered to the graph determined by pairs of quasilinear rays–apparently a fundamental object in the theory of supertropical quadratic forms.

Original language | English |
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Pages (from-to) | 2347-2389 |

Number of pages | 43 |

Journal | Linear and Multilinear Algebra |

Volume | 68 |

Issue number | 12 |

DOIs | |

State | Published - 1 Dec 2020 |

Externally published | Yes |

## Keywords

- Cauchy-Schwarz ratio
- J. Draisma
- Tropical algebra
- bilinear forms
- convex sets
- quadratic forms
- quadratic pairs
- quasilinear sets
- ray spaces
- supertropical modules