Tropical varieties with polynomial weights and corner loci of piecewise polynomials
Alexander Esterov

TL;DR
This paper extends the combinatorial tools for tropical geometry to arbitrary polytopes, providing a new explicit formula for mixed volumes and exploring the structure of tropical varieties with polynomial weights.
Contribution
It introduces an extension of the cohomology models for toric varieties to arbitrary polytopes, leading to a new formula for mixed volumes and insights into tropical subvarieties.
Findings
Derived an explicit formula for mixed volume using support functions.
Extended combinatorial models to non-rational polytopes.
Showed local representation of tropical subvarieties as intersections with possibly negative weights.
Abstract
We find a relation between mixed volumes of several polytopes and the convex hull of their union, deducing it from the following fact: the mixed volume of a collection of polytopes only depends on the product of their support functions (rather than on the individual support functions). For integer polytopes, this dependence is essentially a certain specialization of the isomorphism between two well-known combinatorial models for the cohomology of toric varieties, however, this construction has not been extended to arbitrary polytopes so far (partially due to the lack of combinatorial tools capable of substituting for toric geometry, when vertices are not rational). We provide such an extension, which leads to an explicit formula for the mixed volume in terms of the product of support functions, and may also be interesting because of the combinatorial tools (tropical varieties with…
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