ISSN:
0025-5874
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract. 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, \[ z^2=x^2+(-1)^ky^k,k\ge3,\quad\mbox{or}\quad z^2=yx^2+(-1)^ky^k,k\ge4. \] In particular, except for the stems $z^2=x^2$ and $z^2=yx^2$ , all singularities are real rational double points.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/PL00004692
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