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First steps in tropical geometry. (English) Zbl 1093.14080
Litvinov, G. L. (ed.) et al., Idempotent mathematics and mathematical physics. Proceedings of the international workshop, Vienna, Austria, February 3–10, 2003. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3538-6/pbk). Contemporary Mathematics 377, 289-317 (2005).
The authors give an introduction to tropical geometry, with an emphasis on the geometry in the tropical projective plane. The additive group \((\mathbb{R},+)\) together with the binary operation “min” is a semiring, called the tropical semiring. The tropical semiring is used to build a theory that has many analogies with classical complex algebraic geometry – tropical geometry. The min-operation takes the role of classical addition; the classical addition serves as tropical multiplication. The geometric objects of tropical geometry, called tropical varieties, are polyhedral cell complexes that are contained in tropical affine spaces or in tropical projective spaces. The field \(K\) of Puiseux series with coefficients in \(\mathbb{C}\) carries the order valuation, which takes its values in the tropical semiring. The order valuation is used to associate a tropical variety with every algebraic variety \(V(I)\subseteq(K\setminus \{0\})^n\). Tropical hypersurfaces are described using tropical polynomials. There is a Bézout theorem for the intersection of tropical curves. The quadratic tropical curves through four given points of the plane are described completely. The classical incidence theorem of Pappus does not carry over to tropical geometry. However, there is a constructive version instead.
For the entire collection see [Zbl 1069.00011].

14P99 Real algebraic and real-analytic geometry
52B70 Polyhedral manifolds
16Y60 Semirings
12J10 Valued fields
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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