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Sums of two squares in analytic rings

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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

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


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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
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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