Zeros of discriminants constructed from Hermite–Pade polynomials of an algebraic function and their relation to branch points

Let $f_\infty$ be the germ at $\infty$ of some algebraic function $f$ of degree $m+1$. Let $Q_{n,j}$, $j=0,…,m$, be the Hermite–Pade polynomials of the first type of order $n\in\mathbb N$ constructed from the tuple of germs $[1, f_ \infty, f_\infty^2,…,f_\infty^m]$. We study the asymptotic properties of discriminants constructed from the Hermite–Pade polynomials in question, that is, the discriminants $D_n(z)$ of the polynomials $Q_{n,m}(z)w^m+Q_{n,m-1}(z)w^{m-1}+…+Q_{n,0}(z)$. We find their weak asymptotics, as well as the asymptotic behaviour of their ratio with the polynomial $Q_{n,m}^{2m-2}$. In addition, we refine the weak asymptotic formulae for $D_n$ at branch points of the original algebraic function $f$ and apply the results obtained to the problem of finding branch points of $f$ numerically on the basis of the prescribed germ $f_\infty$, which is used in applied problems. Bibliography: 49 titles.

Hermite–Pade polynomials, discriminants, branch points, algebraic functions, weak asymptotics