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Sums of two squares in analytic rings Ruiz Sancho, Jesús María 512.7 510.22 514.12 515.171.5 Sums of two squares analytic rings Brieskorn singularity complete intersecton Geometria algebraica Teoría de conjuntos 1201.01 Geometría Algebraica 1201.02 Teoría Axiomática de Conjuntos 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. DGICYT Depto. de Álgebra, Geometría y Topología Fac. de Ciencias Matemáticas TRUE pub 2023-06-20T17:11:39Z 2023-06-20T17:11:39Z 1999-02 journal article https://hdl.handle.net/20.500.14352/57922 0025-5874 10.1007/PL00004692 eng PB95-0354 restricted access application/pdf Springer