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Non-archimedean amoebas and tropical varieties. (English) Zbl 1115.14051

This is a remarkable foundational work in tropical algebraic geometry. The non-archimedean amoeba of an algebraic variety in the torus over an algebraically closed field with a non-archimedean valuation is the (closure of the) valuation image of the variety. In case of a hypersurface, the amoeba is the non-differentiability locus of a convex piecewise linear function (so-called “Kapranov theorem”, known from 2000). In the general case, the authors prove that the amoeba coincides with the Bieri-Groves polyhedral set, the latter being the union of the images of all the valuations of the coordinate ring of the given algebraic variety, which extend the valuation of the ground field. Moreover, it is proven that the amoeba of an irreducible variety of dimension \(r\) is a connected polyhedral complex of pure dimension \(r\). In the proofs the authors use the techniques of non-archimedean analysis, in particular, B. Conrad’s theorem on the connectedness of the analytic space of an irreducible algebraic variety [Ann. Inst. Fourier 49, No. 2, 473–541 (1999; Zbl 0928.32011)]. As application, the authors describe the nonexpansive set for an algebraic \({\mathbb Z}^d\)-action on a connected compact group as the radial projection of the adelic amoeba of the annihilator in the module dual to the given group.

MSC:

14T05 Tropical geometry (MSC2010)
14P05 Real algebraic sets
14G20 Local ground fields in algebraic geometry
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
32A60 Zero sets of holomorphic functions of several complex variables
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
28D20 Entropy and other invariants

Citations:

Zbl 0928.32011
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References:

[1] Douglas Lind, Soc and Klaus Schmidt Homoclinic points of algebraic Zd - actions Amer Math Soc Lind Klaus Schmidt and Thomas Ward Mahler measure and entropy for commuting automor - phisms of compact groups Invent, Math 12 pp 129– (2001)
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