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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.

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