On amoebas of algebraic sets of higher codimension
type: Item
instance of: scholarly article
title: On amoebas of algebraic sets of higher codimension
zbMATH Open document ID: 1312.14135
DOI: 10.1134/S0081543812080056
author: N. A. Buruchenko
author: Avgust K. Tsikh
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2014-10-09T00:00:00Z
review text: For any complex affine variety \(V\) one can define amoeba of \(V\) as a set \[ A(V)=\mathrm{Log}(V)\subset \mathbb R^n, \] where \(\mathrm {Log}(z)=(\log |z_1|,\ldots, \log |z_n|)\). Usually, the complement of amoeba \(\mathbb R^n\setminus A\) has a nicer behavior then the amoeba \(A\), and it is more convenient to describe its properties. The the paper studies the topology of \(\mathbb R^n\setminus A(V)\) for \(V\) complete intersection. The main result is the analog of Lefschetz hyperplane theorem and says that the natural map \[ H_{k-1}(\pi\cap \mathbb R^n\setminus A(V)) \to H_{k-1}(\mathbb R^n\setminus A(V)) \] is injective for \(V\) complete intersection of codimension \(k\) and \(\pi\subset \mathbb R^n\) a hyperplane. It allows to describe the higher homology of the complement of the initial complex variety \(\mathbb C^n\setminus V\). Also the paper studies relation between boundary of the amoeba (critical points of the \(\mathrm{Log}\) map above) and logarithmic Gauss map for algebraic sets.
Mathematics Subject Classification ID: 14P25
Mathematics Subject Classification ID: 14M10
Mathematics Subject Classification ID: 32C99
Mathematics Subject Classification ID: 14T05
zbMATH DE Number: 6353216
zbMATH Keywords: amoebas
zbMATH Keywords: complete intersection
zbMATH Keywords: logarithmic Gauss map
MaRDI profile type: Publication
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full work available at URL: s0081543812080056
OpenAlex ID: W2041600451
description: scientific article; zbMATH DE number 6353216