736 research outputs found
A tropical characterization of complex analytic varieties to be algebraic
In this paper we study a -dimensional analytic subvariety of the complex
algebraic torus. We show that if its logarithmic limit set is a finite rational
-dimensional spherical polyhedron, then each irreducible component of
the variety is algebraic. This gives a converse of a theorem of Bieri and
Groves and generalizes a result proven in \cite{MN2-11}. More precisely, if the
dimension of the ambient space is at least twice of the dimension of the
generic analytic subvariety, then these properties are equivalent to the volume
of the amoeba of the subvariety being finite.Comment: 7 pages, 3 figure
Higher convexity of coamoeba complements
We show that the complement of the coamoeba of a variety of codimension k+1
is k-convex, in the sense of Gromov and Henriques. This generalizes a result of
Nisse for hypersurface coamoebas. We use this to show that the complement of
the nonarchimedean coamoeba of a variety of codimension k+1 is k-convex.Comment: 14 pages, 5 color figures, minor revision
Analytic varieties with finite volume amoebas are algebraic
In this paper, we study the amoeba volume of a given dimensional generic
analytic variety of the complex algebraic torus (\C^*)^n. When , we show that is algebraic if and only if the volume of its amoeba is
finite. In this precise case, we establish a comparison theorem for the volume
of the amoeba and the coamoeba. Examples and applications to the linear
spaces will be given.Comment: 13 pages, 2 figure
Higher convexity for complements of tropical varieties
We consider Gromov's homological higher convexity for complements of tropical
varieties, establishing it for complements of tropical hypersurfaces and
curves, and for nonarchimedean amoebas of varieties that are complete
intersections over the field of complex Puiseaux series. Based on these
results, we conjecture that the complement of a tropical variety has this
higher convexity, and we prove a weak form of our conjecture for the
nonarchimedean amoeba of a variety over the complex Puiseaux field. One of our
main tools is Jonsson's limit theorem for tropical varieties.Comment: 11 page
Computing metric hulls in graphs
We prove that, given a closure function the smallest preimage of a closed set
can be calculated in polynomial time in the number of closed sets. This
confirms a conjecture of Albenque and Knauer and implies that there is a
polynomial time algorithm to compute the convex hull-number of a graph, when
all its convex subgraphs are given as input. We then show that computing if the
smallest preimage of a closed set is logarithmic in the size of the ground set
is LOGSNP-complete if only the ground set is given. A special instance of this
problem is computing the dimension of a poset given its linear extension graph,
that was conjectured to be in P.
The intent to show that the latter problem is LOGSNP-complete leads to
several interesting questions and to the definition of the isometric hull,
i.e., a smallest isometric subgraph containing a given set of vertices .
While for an isometric hull is just a shortest path, we show that
computing the isometric hull of a set of vertices is NP-complete even if
. Finally, we consider the problem of computing the isometric
hull-number of a graph and show that computing it is complete.Comment: 13 pages, 3 figure
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