Non-archimedean amoebas and tropical varieties

By Manfred Einsiedler, M. Kapranov, D. Lind. 2004

355 citations recorded by the source corpus.

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 negative of 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 non-expansive set for a related dynamical system.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/cedb377940cb0f2d82f3552a857f92725a1810b4

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Citations in the corpus

The Bergman complex of a matroid and phylogenetic trees (2003, background): “They are the non-archimedean amoebas of [7] and the tropical varieties of [13,15].”
Tropical analytic geometry, Newton polygons, and tropical intersections (2010, methodology): “hen the intersection is not tropically proper. These results are naturally formulated and proved using the theory of tropicalizationsof rigid-analytic spaces, as introduced by Einsiedler-Kapranov-Lind[EKL06] and Gubler [Gub07b]. We have written this paper to be as readable as possible both to tropic”
ELIMINATION THEORY FOR TROPICAL VARIETIES (2007, background): “3 (Einsiedler–Kapranov–Lind [7]).”
A Field of Generalised Puiseux Series for Tropical Geometry (2007, background): “8 ·e),(2,1) ,(3,π) ,(4,0) ,(5,0) ,(6,1)} ,”
Linear independence over tropical semirings and beyond (2008, background): “... structures have appeared in relation with in a variety of fields, like operations research and optimization [Vor67, CG79, GM08b], discrete event systems [BCOQ92, CGQ99], automata theory (see [Pin98] and its references), quasi-classic asymptotics [Mas87], Hamilton-Jacobi partial differential equa”
Super currents and tropical geometry (2010, background): “([4]) A tropical variety is given by the closure of V al({G = 0}) in R, where G ∈ K[z1, .”
Real Tropicalization and Analytification of Semialgebraic Sets (2018, methodology): “\R ˙: If Sis closed, then all these agree with the closure of trop r (S L) in Rn. This is an analogue of the Fundamental Theorem of Tropical Geometry for semialgebraic sets; see [Dra08, Theorem 4.2], [EKL06], and [MS15, x3.2]. We will see later in Remarks6.7that it is necessary to deal with each ort”
Tropical Geometry and its applications (2006, neutral): “The semifield T is closely related to non-Archimedean fields K with (real) valuations val : K → T.”
The tropical momentum map: a classification of toric log symplectic manifolds (2014, background): “19 The notion of tropical domain defined above is closely related to the notion of extended tropicalization Trop(X ) of a complex toric variety X introduced by Kajiwara [10] and Payne [17] (see also [5] for similar ideas).”
Tropical Cramer Determinants Revisited (2013, background): “This is actually a special case of a result of Kapranov characterizing the non-Archimedean amoeba of a hypersurface [EKL06].”
Tropical Discriminants (2005, neutral): “Dedicated to the memory of Pilar Pisón Casares”
A tropical approach to secant dimensions (2006, methodology): “A G ] 5 S ep 2”
Logarithmic limit sets of real semi-algebraic sets (2007, background): “er [Be], and then they were further studied by Bieri and Groves in [BG]. Recently their relations with the theory of non-archimedean ﬁelds and tropical geometry were discovered (see for example [SS], [EKL] and [BJSST]). For these reasons they are often called Bergman fans, Bergman sets, Bieri-Groves”