%0 Journal Article
%A Ruiz Sancho, Jesús María
%T Sums of two squares in analytic rings
%D 1999
%@ 0025-5874
%U https://hdl.handle.net/20.500.14352/57922
%X 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.
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