Newton polytopes and tropical geometry

被引：8
作者：
Kazarnovskii, B. Ya. [1 ]
Khovanskii, A. G. [2 ,3 ]
Esterov, A. I. [4 ]
机构：
[1] Russian Acad Sci, Inst Informat Transmiss Problems, Moscow, Russia
[2] Univ Moscow, Moscow, Russia
[3] Univ Toronto, Toronto, ON, Canada
[4] Natl Res Univ Higher Sch Econ, Moscow, Russia
来源：
RUSSIAN MATHEMATICAL SURVEYS | 2021年 / 76卷 / 01期
基金：
俄罗斯基础研究基金会;
关键词：
family of algebraic varieties; Newton polytope; ring of conditions; toric variety; tropical geometry; mixed volume; exponential sum; ALGEBRAIC-CURVES; VARIETIES; POLYHEDRA;
D O I：
10.1070/RM9937
中图分类号：
O1 [数学];
学科分类号：
0701 ; 070101 ;
摘要：
The practice of bringing together the concepts of 'Newton polytopes', 'toric varieties', 'tropical geometry', and 'Grobner 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.
引用
收藏
页码：91 / 175
页数：85
相关论文
共 50 条
[21]   Nonrational polytopes and fans in toric geometry [J].
Battaglia, Fiammetta ;
Prato, Elisa .
RIVISTA DI MATEMATICA DELLA UNIVERSITA DI PARMA, 2023, 14 (01) :67-86
[22]   Approximating the volume of tropical polytopes is difficult [J].
Gaubert, Stephane ;
MacCaig, Marie .
INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION, 2019, 29 (02) :357-389
[23]   Pure dimension and projectivity of tropical polytopes [J].
Izhakian, Zur ;
Johnson, Marianne ;
Kambites, Mark .
ADVANCES IN MATHEMATICS, 2016, 303 :1236-1263
[24]   Coercive polynomials: stability, order of growth, and Newton polytopes [J].
Bajbar, Tomas ;
Stein, Oliver .
OPTIMIZATION, 2019, 68 (01) :99-124
[25]   On basic concepts of tropical geometry [J].
O. Ya. Viro .
Proceedings of the Steklov Institute of Mathematics, 2011, 273 :252-282
[26]   Newton polytopes of two-dimensional hidden Markov models [J].
Wierer, Jay ;
Boston, Nigel .
EXPERIMENTAL MATHEMATICS, 2007, 16 (02) :227-237
[27]   HOMOTOPIES EXPLOITING NEWTON POLYTOPES FOR SOLVING SPARSE POLYNOMIAL SYSTEMS [J].
VERSCHELDE, J ;
VERLINDEN, P ;
COOLS, R .
SIAM JOURNAL ON NUMERICAL ANALYSIS, 1994, 31 (03) :915-930
[28]   Tropical geometry and the motivic nearby fiber [J].
Katz, Eric ;
Stapledon, Alan .
COMPOSITIO MATHEMATICA, 2012, 148 (01) :269-294
[29]   On globally diffeomorphic polynomial maps via Newton polytopes and circuit numbers [J].
Bajbar, Tomas ;
Stein, Oliver .
MATHEMATISCHE ZEITSCHRIFT, 2018, 288 (3-4) :915-933
[30]   On globally diffeomorphic polynomial maps via Newton polytopes and circuit numbers [J].
Tomáš Bajbar ;
Oliver Stein .
Mathematische Zeitschrift, 2018, 288 :915-933
12345