Nash resolution for binomial varieties as Euclidean division. A priori termination bound, polynomial complexity in essential dimension 2
type: Item
instance of: scholarly article
title: Nash resolution for binomial varieties as Euclidean division. A priori termination bound, polynomial complexity in essential dimension 2
zbMATH Open document ID: 1287.14004
author: Pierre Milman
author: Dima Grigoriev
published in: Advances in Mathematics
publication date: 2012-11-20T00:00:00Z
review text: The authors establish for affine binomial varieties of dimension 2 an a priori bound on the lenght of resolution of singularities by means of the compositions of the normalizations with Nash blowings up. For this purpose they prove a structure theorem for binomial varieties, i.e. isomorphisms of the irreducible components of any binomial variety with its toric component. Then they establish the equivalence of Nash blowings up to an Euclidean \(m\)-dimensional division. Finally a combinatorial version of the composition of normalizations with Nash blowings up for \(m=2\) yields a sharp a priori bound and the algorithm is of a very small complexity. In an Appendix by M. Spivakovsky a bound on the length of the normalized Nash resolution of a minimal singularity via the size of the dual graph of its minimal desingularization is given. The reviewer means that it would be interesting to search for a connection with Hirzebruch-Jung continued fractions and also with Gröbner fans.
reviewed by: Heinrich Reitberger
Mathematics Subject Classification ID: 14E15
Mathematics Subject Classification ID: 52C45
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 68Q25
zbMATH DE Number: 6108933
zbMATH Keywords: Nash blowings up
zbMATH Keywords: binomial varieties
zbMATH Keywords: desingularization
zbMATH Keywords: a priori bound
zbMATH Keywords: multidimensional Euclidean algorithm
zbMATH Keywords: polynomial complexity
MaRDI profile type: Publication
full work available at URL: j.aim.2012.08.009
OpenAlex ID: W2027646475
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DOI: 10.1016/J.AIM.2012.08.009
description: scientific article; zbMATH DE number 6108933