Let $X\subset {{ℂ}^n}^n$ be a closed real-analytic subset and put$$𝒜:=\{z\in X\mid \exists A\subset X,\text{germ}\text{of}\text{a}\text{complex-analytic}\text{set,}z\in A,{dim}_{z}A\>0\}$$This article deals with the question of the structure of $𝒜$. In the main result a natural proof is given for the fact, that $𝒜$ always is closed. As a main tool an interesting relation between complex analytic subsets of $X$ of positive dimension and the Segre varieties of $X$ is proved and exploited.

This article continues the investigation of the analytic intersection algorithm from the perspective of deformation to the normal cone, carried out in the previous papers of the author [7, 8, 9]. The main theorem asserts that, given an analytic set V and a linear subspace S, every collection of hyperplanes, admissible with respect to an algebraic bicone B, realizes the generalized intersection index of V and S. This result is important because the conditions for a collection of hyperplanes to be...