2004/08/31 by Manfred Einsiedler, Mikhail Kapranov, Douglas Lind · 2 citations
Mathematics · #math.AG #math.AC #math.DS #msc:14J29 #msc:13A18 #msc:13C15 #msc:13E15
published as J. Reine Angew. Math. 601 (2006), 139-157 · 19 pages, 2 figures. Added AIM preprint number
arxiv created 2005/01/22 · arxiv updated 2015/12/23
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.