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 Mosc. Math. J., 2012, Volume 12, Number 1, Pages 55–76 (Mi mmj448)

Tropical varieties with polynomial weights and corner loci of piecewise polynomials

A. Esterov

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 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 polynomial weights and their corner loci) that appear in our construction. As an example of another possible application of these new objects, we notice that every tropical subvariety in a tropical manifold $M$ can be locally represented as the intersection of $M$ with another tropical variety (possibly with negative weights), and conjecture certain generalizations of this fact to singular $M$. The above fact about subvarieties of a tropical manifold may be of independent interest, because it implies that the intersection theory on a tropical manifold, which was recently constructed by Allerman, Francois, Rau and Shaw, is locally induced from the ambient vector space.

Key words and phrases: tropical variety, mixed volume, matroid fan, piecewise polynomial, corner locus, intersection theory, cohomology, differential ring, toric variety.

DOI: https://doi.org/10.17323/1609-4514-2012-12-1-55-76

Full text: http://www.ams.org/.../abst12-1-2012.html
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MSC: 14T05, 14M25, 52A39
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Citation: A. Esterov, “Tropical varieties with polynomial weights and corner loci of piecewise polynomials”, Mosc. Math. J., 12:1 (2012), 55–76

Citation in format AMSBIB
\Bibitem{Est12} \by A.~Esterov \paper Tropical varieties with polynomial weights and corner loci of piecewise polynomials \jour Mosc. Math.~J. \yr 2012 \vol 12 \issue 1 \pages 55--76 \mathnet{http://mi.mathnet.ru/mmj448} \crossref{https://doi.org/10.17323/1609-4514-2012-12-1-55-76} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2952426} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000309364900005} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. B. Ya. Kazarnovskii, “On the Action of the Complex Monge–Ampère Operator on Piecewise Linear Functions”, Funct. Anal. Appl., 48:1 (2014), 15–23
2. B. Ya. Kazarnovskii, “Action of the complex Monge–Ampère operator on piecewise-linear functions and exponential tropical varieties”, Izv. Math., 78:5 (2014), 902–921
3. Schneider R., “a Formula For Mixed Volumes”, Geometric Aspects of Functional Analysis: Israel Seminar (Gafa) 2011-2013, Lect. Notes Math., Lecture Notes in Mathematics, 2116, eds. Klartag B., Milman E., Springer Int Publishing Ag, 2014, 423–426
4. B. Ya. Kazarnovskii, “On the product of cocycles in a polyhedral complex”, Izv. Math., 81:2 (2017), 329–358
5. W. Gubler, K. Kuennemann, “A tropical approach to nonarchimedean Arakelov geometry”, Algebr. Number Theory, 11:1 (2017), 77–180
6. Esterov A., “Characteristic Classes of Affine Varieties and Plucker Formulas For Affine Morphisms”, J. Eur. Math. Soc., 20:1 (2018), 15–59