Sums of two squares in analytic rings

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 , , or ? 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, In particular, except for the stems and , all singularities are real rational double points. Received April 4, 1997; in final form September 25, 1997