On basic concepts of tropical geometry

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.