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Tropical algebraic geometry. (English) Zbl 1109.14038
This is a survey of an active new field of mathematics, tropical algebraic geometry, which can be viewed as an algebraic geometry over the real max-plus algebra. The tropical varieties appear to be certain polyhedral complexes, the tropical morphisms are piecewise linear maps, and they appear, for example in logarithmic limits of complex algebraic varieties or as valuation images of algebraic varieties over non-Archimedean fields.
The author demonstrates some very general principles to translate algebro-geometric problems into purely combinatorial ones and illustrates this in several examples concerning plane tropical curves, among them tropical degree-genus formula, tropical BĂ©zout theorem, group structure on a tropical elliptic curve, tropical computation of Gromov-Witten and Welschinger invariants of toric surfaces.

MSC:
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
05C99 Graph theory
12J10 Valued fields
12J25 Non-Archimedean valued fields
14A10 Varieties and morphisms
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14P99 Real algebraic and real-analytic geometry
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