Maslov dequantization

Related entries include idempotent semiring, quantization, tropical geometry, amoeba?

  • G. L. Litvinov, The Maslov dequantization, idempotent and tropical mathematics: A brief introduction, math.GM/0507014

An idempotent version of real algebraic geometry was discovered in the report of O. Viro for the Barcelona Congress [172]. Starting from the idempotent correspondence principle O. Viro constructed a piecewise-linear geometry of polyhedra of a special kind in finite dimensional Euclidean spaces as a result of the Maslov dequantization of real algebraic geometry. He indicated important applications in real algebraic geometry (e.g., in the framework of Hilbert’s 16th problem for constructing real algebraic varieties with prescribed properties and parameters) and relations to complex algebraic geometry and amoebas in the sense of I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky (see their book [61] and [173]). Then complex algebraic geometry was dequantized by G. Mikhalkin and the result turned out to be the same; this new ‘idempotent’ (or asymptotic) geometry is now often called the tropical algebraic geometry (…)

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  • Tropical and Idempotent Mathematics: International Workshop TROPICAL-07 …By G. Grigorii Lazarevich Litvinov, S. N. Sergeev gBooks
  • G. L. Litvinov, V.P. Maslov, The correspondence principle for Idempotent Calculus and some computer applications, In book Idempotency J. Gunawardena (Editor), Cambridge University Press, Cambridge, 1998, p.420-443

“ere exists a (heuristic) correspondence, in the spirit of the correspondence principle in Quantum Mechanics, between important, useful and interesting constructions and results over the field of real (or complex) numbers (or the semiring of all nonnegative numbers) and similar constructions and results over idempotent semirings.’‘

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