Let \(B\) be a polynomial of exact degree \(\nu\) with all its zeros in the unit disk \(\mathbb{D}\). Let \(w_1,\dots,w_m\) be distinct points in the exterior of \(\overline{\mathbb{D}}\) and let \(W(z)=\prod^m_{k=1}(z-w_k)\). Let \(f\) be an analytic function on \(\mathbb{D}\) but on no larger concentric disk, let \(f\) be continuous on \(\overline{\mathbb{D}}\) and let \(f\) have no common zero with \(B\). Define \(P/Q\) and \(R/S\) by the Hermite-Padé interpolants to \(f/B\) and \(f\cdot W/B\) respectively in the \((n+\nu-1)\)th roots of unity.
The authors prove that the coefficients of \(z^{n-j}\) \((j=0,\dots,m-1)\) in the expansion of \(P\) vanish if and only if \(R(w_j)=0\), \(j=1,\dots,m\). Thereby, they extend a theorem of K. G. Ivanov and E. B. Saff [Computational methods and function theory, Proc. Conf., Valparaiso/Chile 1989, 81-87 (1990; Zbl 0707.30031)] and they obtain a further result on simultaneous Hermite-Padé interpolation by a finite set of functions.