Publication: Complexity of global semianalytic sets in a real analytic manifold of dimension 2
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2001
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Walter de Gruyter
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Let X subset of R-n be a real analytic manifold of dimension 2. We study the stability index of X, s(X), that is the smallest integer s such that any basic open subset of X can be written using s global analytic functions. We show that s(X) = 2 as it happens in the semialgebraic case. Also, we prove that the Hormander-Lojasiewicz inequality and the Finiteness Theorem hold true in this context. Finally, we compute the stability index for basic closed subsets, S, and the invariants t and (t) over bar for the number of unions of open (resp. closed) basic sets required to describe any open (resp. closed) global semianalytic set.
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