On amoebas of algebraic sets of higher codimension

Bushueva, N. A.; Tsikh, A. K.

doi:10.1134/s0081543812080056

Cited by 12 publications

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“…6.4] for hypersurfaces. This, together with a technical lemma, allows us to use the result of Bushueva and Tsikh [1] to establish our conjecture for the nonarchimedean amoeba of a complete intersection in (K × ) n . The weak form (that the map ι k (1) sends no positive cycle to zero) for any subvariety of (K × ) n also follows by Henriques's result for amoebas.…”

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confidence: 87%

“…Henriques [4] rediscovered this notion and conjectured that the complement of an amoeba of a variety of codimension k+1 in (C × ) n is k-convex, and established a weak form of this conjecture: the map ι k sends no positive class to zero [4]. Bushueva and Tsikh [1] used complex analysis to prove Henriques' conjecture when the variety is a complete intersection. Other than these cases, Henriques' conjecture remains open.…”

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confidence: 99%

“…Now suppose that V is a complete intersection. By Proposition 1.3, we may assume that A t is the amoeba of a complete intersection, and so by the Theorem of Bushueva and Tsikh [1]…”

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confidence: 99%

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Higher convexity for complements of tropical varieties

Nisse¹,

Sottile

2015

Math. Ann.

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Abstract. 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 Puiseux series. Based on these results, we conjecture that the complement of a tropical variety has this higher convexity, and prove a weak form of this conjecture for the nonarchimedean amoeba of any variety over the complex Puiseux field. One of our main tools is Jonsson's limit theorem for tropical varieties.A tropical hypersurface is a polyhedral complex in R n of pure dimension n−1 that is dual to a regular subdivision of a finite set of integer vectors. This implies that every connected component of its complement is convex. A classical (archimedean) amoeba of a complex hypersurface also has the property that every connected component of its complement is convex [2, Ch. 6, Cor. 1.6].Gromov [3, § ] introduced higher convexity. A subset X ⊂ R n is k-convex if for all affine planes L of dimension k+1, the natural map on kth reduced homologyis an injection. Connected and 0-convex is ordinary convexity. Henriques [4] rediscovered this notion and conjectured that the complement of an amoeba of a variety of codimension k+1 in (C × ) n is k-convex, and established a weak form of this conjecture: the map ι k sends no positive class to zero [4]. Bushueva and Tsikh [1] used complex analysis to prove Henriques' conjecture when the variety is a complete intersection. Other than these cases, Henriques' conjecture remains open.An amoeba is the image in R n of a subvariety V of the torus (C × ) n under the coordinatewise map z → log |z|. Similarly, the coamoeba is the image in (S 1 ) n of V under the coordinatewise argument map. Lifting to the universal cover and taking closure gives the lifted coamoeba in R n . There is also a nonarchimedean coamoeba and a lifted nonarchimedean coamoeba [11]. The complement of either type of lifted coamoeba of a variety of codimension k+1 is k-convex [10], which was proven using tropical geometry.We investigate Gromov's higher convexity for complements of tropical varieties. We show that the complement of a tropical curve in R n is (n−2)-convex. Both this and the convexity of tropical hypersurface complements rely only on some properties of tropical

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Higher convexity for complements of tropical varieties

Nisse¹,

Sottile

2015

Math. Ann.

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“…Bushueva and Tsikh [3] proved Henriques's conjecture when the variety is a complete intersection, but the general case remains open. We first prove a version of Henriques's conjecture for coamoebas.…”

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confidence: 99%

Higher convexity of coamoeba complements

Nisse

Sottile

2015

Bull. London Math. Soc.

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“…For the surfaces of codimension greater than 1 the corresponding expression for the logarithmic Gauss map see in [9].…”

Section: Definition 2 ( [8])mentioning

confidence: 99%

On the Asymptotic of Homological Solutions to Linear Multidimensional Difference Equations

Bushueva¹,

Кузвесов²,

Tsikh³

2014

J. Sib. Fed. Univ., Math. phys.

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Given a linear homogeneous multidimensional difference equation with constant coefficients, we choose a pair (γ, ω), where γ is a homological k-dimensional cycle on the characteristic set of the equation and ω is a holomorphic form of degree k. This pair defines a so called homological solution by the integral over γ of the form ω multiplied by an exponential kernel. A multidimensional variant of Perron's theorem in the class of homological solutions is illustrated by an example of the first order equation.

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On amoebas of algebraic sets of higher codimension

Cited by 12 publications

References 14 publications

Higher convexity for complements of tropical varieties

Higher convexity for complements of tropical varieties

Higher convexity of coamoeba complements

On the Asymptotic of Homological Solutions to Linear Multidimensional Difference Equations

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