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 \(z^2=xy\), \(z^2=yx^2-y^3\), \(z^2= x^3+y^4\) 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, \[ z^2= x^2+(-1)^k y^k,\;k\geq 3,\quad \text{or} \quad z^2=yx^2 +(-1)^ky^k,\;k\geq 4. \] In particular, except for the stems \(z^2=x^2\) and \(z^2=yx^2\), all singularities are real rational double points.
scientific article; zbMATH DE number 1336863