Tropical Ehrhart theory and tropical volume
- PDF / 993,893 Bytes
- 34 Pages / 595.276 x 790.866 pts Page_size
- 96 Downloads / 171 Views
RESEARCH
Tropical Ehrhart theory and tropical volume Georg Loho1 and Matthias Schymura2* * Correspondence:
[email protected] BTU Cottbus-Senftenberg, Platz der Deutschen Einheit 1, 03046 Cottbus, Germany Full list of author information is available at the end of the article
Abstract We introduce a novel intrinsic volume concept in tropical geometry. This is achieved by developing the foundations of a tropical analog of lattice point counting in polytopes. We exhibit the basic properties and compare it to existing measures. Our exposition is complemented by a brief study of arising complexity questions. Keywords: Tropical lattices, Ehrhart theory, Tropical volumes, Tropical polytopes, Tropical semiring, Tropical geometry Mathematics Subject Classification: 14T05, 52C07, 52A38
1 Introduction Tropical geometry is the study of piecewise-linear objects defined over the (max, +)semiring that arises by replacing the classical addition ‘+’ with ‘max’ and multiplication ‘·’ with ‘+.’ While this often focuses on combinatorial properties, see [11,25], we are mainly interested in metric properties. Measuring quantities from tropical geometry turned out to be fruitful for a better understanding of interior point methods for linear programming [2] and principal component analysis of biological data [37]. Moreover, it has interesting connections with representation theory [29,38] and computational complexity [22]. Driven by this motivation, we develop a new definition of a volume for tropical convex sets by a thorough investigation of the tropical analog of lattice point counting. This continues the investigation of intrinsic tropical metric properties that started around a tropical isodiametric inequality [15] and tropical Voronoi diagrams [14]. Tropical polytopes are finitely generated tropical convex sets, see (2) in Sect. 2.1. Former work only considered the lattice points Zd in a d-dimensional polytope, see in particular [12]. This idea was used to measure its Euclidean volume and deduce the hardness to compute it by counting the integer lattice points [22]. These lattice points arise naturally through the representation of affine buildings as tropical polytopes [29]. However, we are more interested in lattice points which are conformal with the semiring structure. Varying the semiring as explained in Sect. 2.3 leads to two natural notions: integer lattice points in polytopes over the (max, ·)-semiring and their image under a logarithm map over the (max, +)-semiring. This is related to the concept arising from ‘dequantization,’ but we show in Sect. 4.3 how our tropical volume concept differs from the existing ones [15].
123
© The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material
Data Loading...
