Analytic surface germs with minimal Pythagoras number

Abstract

We determine all complete intersection surface germs whose Pythagoras number is 2, and find that they are all embedded in R-3 and have the property that every positive semidefinite analytic function germ is a sum of squares of analytic function germs. In addition, we discuss completely these properties for mixed surface germs in R-3. Finally, we find in higher embedding dimension three different families with these same properties.