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Isolated intersection multiplicity and regular separation of analytic sets. (English) Zbl 0784.32005
Let $$X,Y$$ be analytic subsets of an $$m$$-dimensional normed complex vector space $$M$$. Let $$a$$ be an isolated point of $$X \cap Y$$ and $$p>0$$. We say that $$X$$ and $$Y$$ are $$p$$-separated at $$a$$ if $\rho(z,X)+\rho(z,Y) \geq c \rho(z,X \cap Y)^ p$ for $$z$$ in a neighbourhood of $$a$$, for some $$c>0$$ $$(\rho(\cdot,Z)$$ denotes the distance function to the set $$Z \subset M)$$. Consider $$P:=\{p>0$$: $$X$$ and $$Y$$ are $$p$$-separated at $$a$$} and $$p_ 0:= \inf P$$.
The author proves that if $$X,Y$$ are pure-dimensional, then
1. $$p_ 0 \in P \cap \mathbb{Q}$$,
2. $$1 \leq p_ 0 \leq i(X \cdot Y;a)-\deg_ aX \deg_ aY+1$$,
where $$i(X \cdot Y;a)$$ is the multiplicity of improper isolated intersection of $$X$$ and $$Y$$ [R. Achilles, the author and T. Winiarski, ibid. 51, 21-36 (1990)].
Moreover, he generalizes this result (in the same formulation) to manifolds (instead of $$M)$$ and non pure-dimensional case.

MSC:
 32B10 Germs of analytic sets, local parametrization 14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
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