Conjectured asymptotics for type II Hermite–Padé polynomials on a three-sheeted Riemann surface

Let $f\in\mathbb C(z,w)$, let $w_\infty$ be the specified element of the function $w$, and let $f_\infty\in\mathscr H(\infty)$ be the corresponding element of $f$. For each $n\in\mathbb N$, let $P_{2n},P_{2n,1},P_{2n,2}$ be the corresponding type II Hermite–Padé polynomials for $f$ and $f^2$ with multiindex $(2n,2n)$, with degrees at most $2n$. Here $\chi(P)$ denotes the normalized zero-counting measure, $\lambda_E$ is the equilibrium measure on $E$, $D$ is the relevant domain, and $z^{(0)}$ denotes the distinguished sheet above $z$. Asymptotic Hermite–Padé conjecture. If $f\in\mathbb C(z,w)$ and $f_\infty\in\mathscr H(\infty)$ corresponds to the specified $w_\infty$, then, as $n\to\infty$,

$$\frac1n\chi(P_{2n}),\quad\frac1n\chi(P_{2n,1}),\quad\frac1n\chi(P_{2n,2})\overset{*}{\longrightarrow}\lambda_E,$$

while

$$\frac{P_{2n,1}}{P_{2n}}(z)\overset{\operatorname{cap}}{\longrightarrow}f(z^{(0)}),\qquad \frac{P_{2n,2}}{P_{2n}}(z)\overset{\operatorname{cap}}{\longrightarrow}f^2(z^{(0)})\quad\text{inside }D.$$

This is presented as a natural conjectural extension of the paper’s asymptotic results for type II Hermite–Padé polynomials. The source does not indicate that the assertion has been proved or refuted.