Idempotent semigroups and tropical algebraic sets. (English) Zbl 1273.14133
This paper deals with tropical algebraic sets with the structure of an idempotent semigroup and their topology. After general results about the topology of idempotent semigroups, the focus is on intersections of tropical hypersurfaces which are called tropical algebraic sets. If a tropical algebraic set is a subsemigroup of \(\mathbb{R}^n\) with the tropical addition, it is called an additive tropical set. The support of a weighted balanced polyhedral complex is called a tropical set-variety. A tropical polynomial is called simple if its monomials are univariate or constant. The authors prove that any tropical algebraic set defined by simple polynomials is additive and conjecture the reverse statement. They prove their conjecture for hypersurfaces, tropical algebraic sets defined by binomials (i.e. usual affine subspaces), plane curves and spatial curves. Finally, they show that the skeletons of an additive tropical set-variety (where the polyhedral structure is naturally defined) are additive, thus their connected componenents are contractible using the general topological results.
Reviewer: Hannah Markwig (Göttingen)
MSC:
14T05 | Tropical geometry (MSC2010) |
20M14 | Commutative semigroups |
06F05 | Ordered semigroups and monoids |
12K10 | Semifields |
13B25 | Polynomials over commutative rings |
22A15 | Structure of topological semigroups |
51M20 | Polyhedra and polytopes; regular figures, division of spaces |
Keywords:
tropical geometry; polyhedral complexes; tropical polynomials; idempotent semigroups; simple polynomialsReferences:
[1] | Allermann, L., Rau, J.: First steps in tropical intersection theory. Math. Z. 264, 633-670 (2010) · Zbl 1193.14074 · doi:10.1007/s00209-009-0483-1 |
[2] | Einsiedler, M., Kapranov, M., Lind, D.: Non-archimedean amoebas and tropical varieties. J. Reine Angew. Math. 601, 139-157 (2006) · Zbl 1115.14051 · doi:10.1515/CRELLE.2006.097 |
[3] | Gathmann, A.: Tropical algebraic geometry. Jahresber. Deutsch. Math.-Verein. 108, 3-32 (2006) · Zbl 1109.14038 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.