Geodesic
On basic concepts of tropical geometry
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mathematics, Tome 273 (2011), pp. 271-303

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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. 
@article{TM_2011_273_a12,
     author = {O. Ya. Viro},
     title = {On basic concepts of tropical geometry},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {271--303},
     publisher = {mathdoc},
     volume = {273},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2011_273_a12/}
}
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O. Ya. Viro. On basic concepts of tropical geometry. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mathematics, Tome 273 (2011), pp. 271-303. http://geodesic.mathdoc.fr/item/TM_2011_273_a12/