On basic concepts of tropical geometry |
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Authors: | O Ya Viro |
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Institution: | 1.Mathematics Department,Stony Brook University,Stony Brook,USA;2.St. Petersburg Department of the Steklov Mathematical Institute,Russian Academy of Sciences,St. Petersburg,Russia |
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Abstract: | 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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Keywords: | |
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