Tropical Ehrhart theory and tropical volume
Georg Loho1, Matthias Schymura2
1London School of Economics and Political Science, Houghton Street, London, WC2A 2AE UK.
Research in the Mathematical Sciences
|October 8, 2020
Summary
Researchers developed a new intrinsic volume concept in tropical geometry, inspired by lattice point counting in polytopes. This novel measure offers a unique perspective on geometric properties within this mathematical field.
Area of Science:
- Tropical geometry
- Discrete geometry
- Algebraic geometry
Background:
- Existing measures in tropical geometry lack a direct analog to classical intrinsic volumes.
- Lattice point counting in polytopes is a fundamental concept in discrete and convex geometry.
Purpose of the Study:
- Introduce a novel intrinsic volume concept within tropical geometry.
- Establish the foundational principles for a tropical analog of lattice point counting.
- Compare the properties of this new measure with existing ones.
Main Methods:
- Development of a tropical analog for lattice point counting in polytopes.
- Theoretical exposition of the properties of the novel intrinsic volume.
- Comparative analysis against established geometric measures.
Main Results:
- Introduction of a new intrinsic volume concept tailored for tropical geometry.
- Demonstration of the basic properties and behaviors of this novel measure.
- Initial exploration of computational complexity associated with the new concept.
Conclusions:
- The proposed intrinsic volume offers a new tool for tropical geometry.
- This work bridges concepts from discrete geometry and tropical algebraic geometry.
- Further research into the complexity and applications of this tropical intrinsic volume is warranted.
Keywords:
Ehrhart theoryTropical geometryTropical latticesTropical polytopesTropical semiringTropical volumesMore Related Videos
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