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...C issn proceedings of the steklov institute mathematics vol pp pleiades publishing ltd original russian text o ya viro published in trudy matematicheskogo instituta imeni v a steklova on basic concepts tropical geometry viroa b received april abstract we introduce binary operation over complex numbers that is analog addition this together with ordinary multiplication satis es axioms generalize standard eld algebraic hypereld thus dened occupies an intermediate position between classical com plex and deformation similar to litvinov maslov dequantization real leads degeneration varieties into whereas amoeba variety turns out be corresponding modications multivalued additions are constructed for other elds as well p adic quaternions doi s introduction when one wishes describe by single phrase says it semield t rmax elements augmented role played taking maximum two max zero whiletheroleofunity r properties valid following exception completely noninvertible i e any there does not exist x su...