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Tropical ideals do not realise all Bergman fans
Jan Draisma1,2, Felipe Rincón3
1Universität Bern, Mathematisches Institut, Sidlerstrasse 5, 3012 Bern, Switzerland.
Research in the Mathematical Sciences
|November 15, 2021
Summary
Tropical geometry explores tropical varieties, which are weighted polyhedral complexes. This study proves that certain weighted complexes, specifically the Bergman fan of a direct sum of matroids, cannot be realized as tropical varieties.
Area of Science:
- Tropical geometry
- Algebraic geometry
- Combinatorial geometry
Background:
- Tropical varieties are finite polyhedral complexes with positive integral weights on maximal cells, associated with tropical ideals.
- The realizability question in tropical geometry asks which weighted polyhedral complexes can be realized as tropical varieties.
Purpose of the Study:
- To investigate the realizability of specific weighted polyhedral complexes as tropical varieties.
- To determine if the Bergman fan of the direct sum of the Vámos matroid and the uniform matroid (U(2,3)) with unit weights can be a tropical variety.
Main Methods:
- Utilizing the work of Las Vergnas on the non-existence of tensor products of matroids.
- Applying combinatorial and algebraic techniques from tropical geometry.
Main Results:
- Proved that there is no tropical ideal whose associated tropical variety is the Bergman fan of the direct sum of the Vámos matroid and the uniform matroid U(2,3).
- Demonstrated that this specific complex, with all maximal cones having weight one, cannot be realized as a tropical variety.
Conclusions:
- The study provides a negative answer to a specific realizability question in tropical geometry.
- Highlights the limitations and constraints in constructing tropical varieties from certain combinatorial structures like direct sums of matroids.
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