Given a semianalytic set S in a complex space and a point p in S, there is a
unique smallest complex-analytic germ at p which contains the germ of S, called
the holomorphic closure of S at p. We show that if S is semialgebraic then its
holomorphic closure is a Nash germ, for every p, and S admits a semialgebraic
filtration by the holomorphic closure dimension. As a consequence, every
semialgebraic subset of a complex vector space admits a semialgebraic
stratification into CR manifolds satisfying a strong version of the condition
of the frontier.Comment: Published versio
Abstract. Given a real analytic set X in a complex manifold and a positive integer d, denote by A d the set of points p in X at which there exists a germ of a complex analytic set of dimension d contained in X. It is proved that A d is a closed semianalytic subset of X.
We prove that if a strongly minimal non-locally modular reduct of an algebraically closed valued field of characteristic 0 contains +, then this reduct is bi-interpretable with the underlying field.
We show that an analogue of the Hilbert's Thirteenth Problem fails in the real subanalytic setting. Namely we prove that, for any integer n, the o-minimal structure generated by restricted analytic functions in n variables is strictly smaller than the structure of all global subanalytic sets, whereas these two structures define the same subsets in R n+1 .
This article presents two constructions motivated by a conjecture of van den Dries and Miller concerning the restricted analytic field with exponentiation. The first construction provides an example of two o-minimal expansions of a real closed field that possess the same field of germs at infinity of onevariable functions and yet define different global one-variable functions. The second construction gives an example of a family of infinitely many distinct maximal polynomially bounded reducts (all this in the sense of definability) of the restricted analytic field with exponentiation.
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