On basic concepts of tropical geometry

被引：0
作者：
O. Ya. Viro
机构：
[1] Stony Brook University,Mathematics Department
[2] Russian Academy of Sciences,St. Petersburg Department of the Steklov Mathematical Institute
来源：
Proceedings of the Steklov Institute of Mathematics | 2011年 / 273卷
关键词：
STEKLOV Institute; Tropical Variety; Tropical Geometry; Newton Polytope; Ordinary Multiplication;
D O I：
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学科分类号：
摘要：
We introduce a binary operation over complex numbers that is a tropical analog of addition. This operation, together with the ordinary multiplication of complex numbers, satisfies axioms that generalize the standard field axioms. The algebraic geometry over a complex tropical hyperfield thus defined occupies an intermediate position between the classical complex algebraic geometry and tropical geometry. A deformation similar to the Litvinov-Maslov dequantization of real numbers leads to the degeneration of complex algebraic varieties into complex tropical varieties, whereas the amoeba of a complex tropical variety turns out to be the corresponding tropical variety. Similar tropical modifications with multivalued additions are constructed for other fields as well: for real numbers, p-adic numbers, and quaternions.
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页码：252 / 282
页数：30
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