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.
Highlights
Tropical geometry is the study of piecewise-linear objects defined over thesemiring 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
1 Introduction Tropical geometry is the study of piecewise-linear objects defined over thesemiring that arises by replacing the classical addition ‘+’ with ‘max’ and multiplication ‘·’ with ‘+.’
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]
Summary
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] 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. 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. 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]
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