System singularity and system pole conjecture for Hermite–Padé approximants

Let ${\bf f}=(f_1,\ldots,f_d)$ and ${\bf m}=(m_1,\ldots,m_d)\in\mathbb{Z}_+^d\setminus\{{\bf 0}\}$, and suppose that the denominator $Q_{n,\bf m}$ is unique for all sufficiently large $n$ and that

$$\lim_{n\to\infty}Q_{n,\bf m}=Q_{|{\bf m}|},\qquad \deg Q_{|{\bf m}|}=|{\bf m}|,6$$

with $Q_{|{\bf m}|}(0)\neq0$. Let $\zeta$ satisfy $Q_{|{\bf m}|}(\zeta)=0$. A system singularity of ${\bf f}$ with respect to ${\bf m}$ is a nonzero complex number at which at least one polynomial combination of the components of ${\bf f}$, analytic on the disk of radius $|\zeta|$, is singular. If such a polynomial combination $F$ determines the system singularity at $\zeta$ and $\zeta\in D_1(F)$, then $\zeta$ is a system pole of ${\bf f}$ with respect to ${\bf m}$ of order equal to the multiplicity of $\zeta$ as a zero of $Q_{|{\bf m}|}$. System singularity and system pole conjecture. Under these assumptions, $\zeta$ is a system singularity of ${\bf f}$ with respect to ${\bf m}$. Moreover, if $\zeta\in D_1(F)$ for some polynomial combination $F$ determining the system singularity at $\zeta$, then $\zeta$ is a system pole of order equal to the multiplicity of $\zeta$ as a zero of $Q_{|{\bf m}|}$. The claim extends the scalar phenomenon relating limiting Padé denominators to singularities and poles, but its validity is not established in the supplied text.