Higher order approximation of complex analytic sets by algebraic sets (Q1756315)

The article concerns local approximation of analytic sets by algebraic sets with any prescribed order of tangency. The main theorem is: Let \(X\) be an analytic subset of some open set in \(\mathbb{C}^{m}\) of pure dimension \(n\) such that \(X\) contains \(0\in \mathbb{C}^{m}\). Then there are an open neighborhood \(U\) of \(0\) in \(\mathbb{C}^{m}\) and a sequence \(\{X_{\nu}\}\) of algebraic subsets of \(\mathbb{C}^{m}\) of pure dimension \(n\) such that \( \{X_{\nu}\cap U\}\) converges to \(X\cap U\) in the sense of chains [\textit{P. Tworzewski}, Ann. Pol. Math. 62, No. 2, 177--191 (1995; Zbl 0911.32018)]. Moreover, \(\text{dist} (X\cap (U\cdot r),X_{\nu}\cap (U\cdot r))\leq r^{\nu}\), for every \(r\in (0,1)\), \(\nu \in \mathbb{N}\). Here \(\text{dist}(\cdot ,\cdot)\) denotes the Hausdorff distance of relatively compact subsets of \(\mathbb{C}^{m}\).