Sums of two squares in analytic rings
type: Item
instance of: scholarly article
title: Sums of two squares in analytic rings
zbMATH Open document ID: 0930.32007
DOI: 10.1007/PL00004692
published in: Mathematische Zeitschrift
publication date: 2000-02-09T00:00:00Z
review text: We study analytic singularities for which every positive semidefinite analytic function is a sum of two squares of analytic functions. This is a basic useful property of the plane, but difficult to check in other cases; in particular, what about \(z^2=xy\), \(z^2=yx^2-y^3\), \(z^2= x^3+y^4\) or \(z^2= x^3-xy^3\)? In fact, the unique positive examples we can find are the Brieskorn singularity, the union of two planes in 3-space and the Whitney umbrella. Conversely, we prove that a complete intersection with that property (other than the seven embedded surfaces already mentioned) must be a very simple deformation of the two latter, namely, \[ z^2= x^2+(-1)^k y^k,\;k\geq 3,\quad \text{or} \quad z^2=yx^2 +(-1)^ky^k,\;k\geq 4. \] In particular, except for the stems \(z^2=x^2\) and \(z^2=yx^2\), all singularities are real rational double points.
Mathematics Subject Classification ID: 32B05
Mathematics Subject Classification ID: 32B10
zbMATH DE Number: 1336863
zbMATH Keywords: sums of two squares
zbMATH Keywords: analytic rings
zbMATH Keywords: Brieskorn singularity
zbMATH Keywords: complete intersecton
author: Jesús M. Ruiz
reviewed by: Jesús M. Ruiz
MaRDI profile type: Publication
full work available at URL: pl00004692
OpenAlex ID: W1973748215
description: scientific article; zbMATH DE number 1336863