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 Ray(V), supported by a ‘supertropical trigonometry’. Employing a (partial) quasiordering on Ray(V), 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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Number of pages | 44 |
Journal | Linear and Multilinear Algebra |
Early online date | 26 Mar 2019 |
DOIs | |
Publication status | E-pub ahead of print - 26 Mar 2019 |
Bibliographical note
Acknowledgements: The authors thank the referee for the helpful suggestions and comments.Keywords
- bilinear forms
- Cauchy-Schwarz ratio
- convex sets
- quadratic forms
- quadratic pairs
- quasilinear sets
- ray spaces
- supertropical modules
- Tropical algebra