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An isolated point of intersection of two analytic sets is considered. We give a sharp estimate of their regular separation exponent in terms of intersection multiplicity and local degrees. 1. Separation. Let M be an m-dimensional normed complex vector space. Following ([4], IV.7) we say that a pair of closed sets X, Y in an open subset G of M satisfies the condition (S) at a point a ∈ G if either a 6∈ X ∩ Y , or a ∈ X ∩ Y and %(z,X) + %(z, Y ) ≥ c%(z,X ∩ Y ) for z in a neighbourhood of a, for some c, p > 0 (%(·, Z) denotes the distance function to the set Z ⊂M). In the sequel we will consider only isolated points of the intersection of X and Y . We say that X and Y are p-separated at a ∈ G if a is an isolated point of X ∩ Y and the pair X,Y satisfies the condition (S) at a, with exponent p and some constant c > 0. As a simple consequence of properties of (S) (see [4], IV.7.1) we get the following lemma. Lemma 1.1. Let H1 ⊂ G and H2 be open subsets of normed , finitedimensional complex vector spaces and let f : H1 → H2 be a biholomorphism. Then closed subsets X and Y of G are p-separated at a point a ∈ H1 if and only if f(X ∩H1) and f(Y ∩H1) are p-separated at f(a). 1991 Mathematics Subject Classification: 32B10, 14C17.

Isolated intersection multiplicity and regular separation of analytic sets