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Ruiz Sancho, Jesús María
(1999)
*Sums of two squares in analytic rings.*
Mathematische Zeitschrift, 230
(2).
pp. 317-328.
ISSN 0025-5874

PDF
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Official URL: http://link.springer.com/content/pdf/10.1007%2FPL00004692

## Abstract

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≥3, or z(2)=yx(2)+(-1)(k)y(k), k≥4. In particular, except for the stems z(2)=x(2) and z(2)=yx(2), all singularities are real rational double points.

Item Type: | Article |
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Uncontrolled Keywords: | Sums of two squares; analytic rings; Brieskorn singularity; complete intersecton |

Subjects: | Sciences > Mathematics > Algebraic geometry Sciences > Mathematics > Set theory |

ID Code: | 19994 |

Deposited On: | 20 Feb 2013 16:26 |

Last Modified: | 12 Dec 2018 15:13 |

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