Ruiz Sancho, Jesús María2023-06-202023-06-201999-020025-587410.1007/PL00004692https://hdl.handle.net/20.500.14352/57922We 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.engjournal articlehttp://link.springer.com/content/pdf/10.1007%2FPL00004692http://link.springer.comrestricted access512.7510.22514.12515.171.5Sums of two squaresanalytic ringsBrieskorn singularitycomplete intersectonGeometria algebraicaTeoría de conjuntos1201.01 Geometría Algebraica1201.02 Teoría Axiomática de Conjuntos