Minimal polynomials for compact sets of the complex plane (Q1357539)

Let \(T_n\) denote the classical Chebyshev polynomial. The author shows that for every complex polynomial \(Wr(z)= a_0+ \cdots+ a_rz^r\), \(a_j \in\mathbb{C}\), there is a set \(A\) such that \((2a_r)^{-n+1} T_n(W_r)\), \(n=1,2, \dots\), is a minimal polynomial on \(A\) and on all equipotential lines for \(CA\) (the complement of \(A)\). As a consequence of this, the author shows that a minimal polynomial on \(l\) disjoint intervals \(E_l\), with maximal number of extremal points on \(E_l\) is also a minimal polynomial on all equipotential lines for \(CE_l\). Such a conclusion implies the corresponding result for two intervals due to \textit{B. Fischer} (Constructive Approximation 8, No. 3, 309-329 (1992; Zbl 0778.41012)].