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Tropical geometry is a relatively new area in mathematics, which might loosely be described as a piece-wise linear or skeletonized version of algebraic geometry. Its leading ideas had appeared in different forms in the earlier works of George M. Bergman and of Robert Bieri and John Groves, but only since the late 1990s has an effort been made to consolidate the basic definitions of the theory. This has been motivated by the applications to enumerative algebraic geometry found by Grigory Mikhalkin. == Basic definitions == ''We will use the ''min convention'', that tropical addition is classical minimum. It is also possible to cast the whole subject in terms of the ''max convention'', negating throughout, and several authors make this choice.'' The ''tropical semiring'' (also known as a ''tropical algebra'' or, with the max convention, the ''max-plus algebra'', due to the name of its operations) is a semiring (ℝ ∪ , ⊕, ⊗), with the operations as follows: : : Tropical exponentiation is defined in the usual way as iterated tropical products (see exponentiation#In abstract algebra). A monomial of variables in this semiring is a linear map, represented in classical arithmetic as a linear function of the variables with integer coefficients.〔David Speyer and Bernd Sturmfels, "Tropical mathematics", ''Mathematics Magazine'' 82:3 (2009), pp. 163–173. (full text )〕 A polynomial in the semiring is the minimum of a finite number of such monomials, and is therefore a concave, continuous, piecewise linear function. The set of points where a tropical polynomial ''F'' is non-differentiable is called its associated tropical hypersurface. There are two important characterizations of these objects: # Tropical hypersurfaces are exactly the rational polyhedral complexes satisfying a "zero-tension" condition.〔 # Tropical surfaces are exactly the non-Archimedean amoebas over an algebraically closed non-Archimedean field ''K''. These two characterizations provide a "dictionary" between combinatorics and algebra. Such a dictionary can be used to take an algebraic problem and solve its easier combinatorial counterpart instead. The tropical hypersurface can be generalized to a tropical variety by taking the non-Archimedean amoeba of ideals ''I'' in ''K''() instead of polynomials. The tropical variety of an ideal ''I'' equals the intersection of the tropical hypersurfaces associated to every polynomial in ''I''. This intersection can be chosen to be finite. There are a number of articles and surveys on tropical geometry. The study of tropical curves (tropical hypersurfaces in ℝ2) is particularly well developed. In fact, for this setting, mathematicians have established analogues of many classical theorems; e.g., Pappus's hexagon theorem, Bézout's theorem, the degree-genus formula, and the group law of the cubics all have tropical counterparts. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Tropical geometry」の詳細全文を読む スポンサード リンク
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