Non-archimedean amoebas and tropical varieties

Abstract

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.

Non-Archimedean Coamoebae

“…This was introduced by Bergman [3], and then Bieri and Groves [4] showed that the cone over it was a rational polyhedral fan. Later work of Kapranov [5], and then Speyer and Sturmfels [19] identified this fan with the negative − Trop(X) of the tropical variety of X, computed in any valued field K containing C with valuation group Γ and residue field C. Here is the solution of these inequalities, the amoeba A (ℓ), and the tropical variety Trop(ℓ). The coamoeba coA (X) of a subscheme X of T N is the image of X(C) under the argument map Arg.…”

Homogeneous coordinate rings and mirror symmetry for toric varieties

“…One may also define amoebas for varieties over fields other than ‫.ރ‬ Amongst other results on varieties over non-Archimedean fields, Kapranov gave a description of their amoebas in term of polyhedral complexes in ‫ޒ‬ n [4]. Mikhalkin then used this description in order to obtain new results about the topology of complex hypersurfaces [15].…”

Constructing polynomial systems with many positive solutions using tropical geometry

“…A consequence of an important result due to Kapranov [10] is that the set T 1 ∩ T 2 contains the image under the valuation map (see Sect. 2.1) of the solutions to (1.4).…”