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Uspekhi Mat. Nauk, 2021, Volume 76, Issue 1(457), Pages 95–190 (Mi umn9937)  

Newton polytopes and tropical geometry

B. Ya. Kazarnovskiia, A. G. Khovanskiibc, A. I. Esterovd

a Institute for Information Transmission Problems of the Russian Academy of Sciences
b Independent University of Moscow
c University of Toronto, Toronto, Canada
d National Research University Higher School of Economics

Abstract: The practice of bringing together the concepts of ‘Newton polytopes’, ‘toric varieties’, ‘tropical geometry’, and ‘Gröbner bases’ has led to the formation of stable and mutually beneficial connections between algebraic geometry and convex geometry. This survey is devoted to the current state of the area of mathematics that describes the interaction and applications of these concepts.
Bibliography: 68 titles.

Keywords: family of algebraic varieties, Newton polytope, ring of conditions, toric variety, tropical geometry, mixed volume, exponential sum.

Funding Agency Grant Number
Canadian Grant 156833-17
Russian Foundation for Basic Research 20-01-00579
The second author was supported by Canadian grant no. 156833-17. The third author was supported by the Russian Foundation for Basic Research (grant no. 20-01-00579).


DOI: https://doi.org/10.4213/rm9937

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English version:
Russian Mathematical Surveys, 2021, 76:1, 91–175

Bibliographic databases:

UDC: 512.7+514.17
MSC: Primary 14M15, 14Txx; Secondary 14C17
Received: 25.11.2019

Citation: B. Ya. Kazarnovskii, A. G. Khovanskii, A. I. Esterov, “Newton polytopes and tropical geometry”, Uspekhi Mat. Nauk, 76:1(457) (2021), 95–190; Russian Math. Surveys, 76:1 (2021), 91–175

Citation in format AMSBIB
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\vol 76
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