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Non-archimedean amoebas and tropical varieties

Manfred Einsiedler·Mikhail Kapranov·Douglas Lind
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Abstract

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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Manfred Einsiedler, Mikhail Kapranov, Douglas Lind. 2004-08-23. Non-archimedean amoebas and tropical varieties. https://arxiv.org/abs/math/0408311

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