The Lojasiewicz structure theorem on p. 169 in the book of Krantz/Parks A primer of real analytic functions confuses me. According to the stratification property, the zeroes of a real analytic function can not be isolated points in e.g. $\mathbb{R}^2$. But what about for example $f(x,y)=x^2+y^2$ where $(0,0)$ is the only zero?
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2$\begingroup$ It looks like a mistake. The sets in the stratification should get larger in dimensions, but some dimensions might get skipped. They seem to require nonempty sets of each dimension. $\endgroup$– Ben McKayMar 16, 2016 at 16:25
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$\begingroup$ I think the dimensions are right. The lower dimensional sets are contained in the closure of the higer dimensional sets. But indeed maybe they require nonempty sets in each dimension. A second possibility is that the rotation (or more general a suitable transformation) in the beginning of the theorem takes in fact place in the complex space, But is this possible? $\endgroup$– user284045Mar 17, 2016 at 11:00
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