Non-archimedean amoebas and tropical varieties

Details Non-archimedean amoebas and tropical varieties Non-archimedean amoebas and tropical varieties

2004-08-23

arxiv.org/abs/math/0408311v2

Mathematics

Algebraic Geometry

19 pages, 2 figures. Added AIM preprint number

Scientific paper

We study the non-archimedean counterpart to the complex amoeba of an algebraic variety, and show that it coincides with a polyhedral set defined by Bieri and Groves using valuations. For hypersurfaces this set is also the tropical variety of the defining polynomial. Using non-archimedean analysis and a recent result of Conrad we prove that the amoeba of an irreducible variety is connected. We introduce the notion of an adelic amoeba for varieties over global fields, and establish a form of the local-global principle for them. This principle is used to explain the calculation of the nonexpansive set for a related dynamical system.

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