This article is cited in 8 papers

Newton polytopes and tropical geometry

B. Ya. Kazarnovskii^a, A. G. Khovanskii^bc, A. I. Esterov<sup>d

a</sup> 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.

UDC: 512.7+514.17

MSC: Primary 14M15, 14Txx; Secondary 14C17

Received: 25.11.2019

DOI: 10.4213/rm9937

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

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