Siciak’s extremal function of non-UPC cusps. I. (English) Zbl 1205.32012
The author studies the \(L\)–regularity of compact subsets of \(\mathbb C^N\) at cusps for which the Pleśniak semianalytic accessibility criterion fails. He presents various applications of the following criterion for the \(L\)-regularity at \(z_0\). There exist \(s_0>0\) and sequences \((d_j)_{j\in\mathbb N}\subset\mathbb N\), \((r_j)_{j\in\mathbb N}\subset[0,s_0)\) such that: \(r_j\to0\), \(d_j\sqrt{r_j}\to0\), and for each \(j\in\mathbb N\) there exists an \(L\)-regular at \(z_0\) compact set \(K_j\subset\mathbb C^N\) such that for any \(z\in K_j\setminus E\) there is a polynomial mapping \(W_z^j:\mathbb C\to\mathbb C^N\) with \(W_z^j(0)=z\), \(\deg W_z^j\leq d_j\), and \(W_z^j([r_j,s_0])\subset E\).
Reviewer: Marek Jarnicki (Kraków)
MSC:
| 32E30 | Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs |
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