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This article is cited in 1 scientific paper (total in 1 paper) Zeros of discriminants constructed from Hermite–Padé polynomials of an algebraic function and their relation to branch points A. V. Komlov, R. V. Palvelev Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
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    § 1. IntroductionLet us formulate the general problem of the recovery of values and properties of an algebraic (or, in a more general setting, multivalued analytic) function from a given germ of it. Let $f$ be an algebraic function of degree $m+1$, that is, the solution of the equation $\mathcal{P}(z,f(z))\equiv 0$ where $\mathcal{P}(z,w)$ is an irreducible polynomial in $z$ and $w$ and $\deg_w \mathcal{P}(z,w)=m+1$. Suppose that we know the Taylor expansion of the germ $f_{z_0}$ of $f$ at some point $z_0$ on the Riemann sphere $\widehat{\mathbb{C}}$. Without loss of generality we assume that $z_0=\infty$, and let $f_\infty$ be holomorphic at $\infty$. So $f_\infty(z) = \sum_{k=0}^\infty c_k z^{-k}$ in a neighbourhood of $\infty$, and the coefficients $c_k$ are known to us, more precisely, we know how to calculate them sequentially. The global problem is to recover values or some analytic properties of $f$ from these Taylor coefficients $c_k$. Furthermore, we wish to do it constructively. We understand ‘constructively’ in the sense of Henrici [18]: this means that the $n$th approximant must be a rational function of a finite number of coefficients $c_k$. In our situation this is not enough, and we also use zeros of polynomials found constructively. The Hermite–Padé polynomials of type I of order $n\in\mathbb{N}$ for the tuple of germs $[1, f_\infty, f_\infty^2,\dots, f_\infty^m]$ at the point $\infty\in\widehat{\mathbb{C}}$ are polynomials $Q_{n,0}, \dots, Q_{n,m}$ such that $\deg{Q_{n,j}}\leqslant{n}$, $j=0, \dots, m$, at least one $Q_{n, j}$ is not identically equal to $0$ and the following relation holds true:
$$
\begin{equation}
Q_{n,0}(z) + \sum_{j=1}^{m}f_\infty^j(z)Q_{n, j}(z)=O\biggl(\frac{1}{z^{m(n+1)}}\biggr) \quad \text{as } z\to\infty.
\end{equation}
\tag{1}
$$
Note that in the setting of the problem we restrict ourselves by the condition that our germ $f_\infty$ is holomorphic at the point $\infty$ just for simplicity of presentation. If we allow $f_\infty$ to have a pole of order $N_0$ at $\infty$, then one should replace the condition $\deg{Q_{n,0}}\leqslant{n}$ in the definition of Hermite–Padé polynomials by the condition $\deg{Q_{n,0}}\leqslant{n+mN_0}$ and keep other conditions the same. All the main results of the paper, presented in § 2, are also true in this more general case, and their proofs repeat fully the ones in the paper, with the corresponding change in sharp constants. In Nuttall’s paper [30] (under some restrictions and not always with full proofs) and in [24] (in full generality) the weak and ratio asymptotics of the Hermite–Padé polynomials $Q_{n,j}$ were found (see Theorem 1 below; it was fully proved in [24] and here we present its formulation just for the completeness of presentation). We emphasize that here, as in [30] and [24], we discuss only the so-called model case when the degree of the algebraic function $f$ equals $m+1$ and one considers the Hermite–Padé polynomials for the tuple of germs $[1, f_\infty, f_\infty^2,\dots, f_\infty^m]$. In [24], § 1, after Theorem 2, it was noted that the zeros $w(z)$ of the polynomial $P_n(z,w)$ obtained by replacing $f_\infty$ by $w$ in the left-hand side of (1),
$$
\begin{equation}
P_n(z,w):=Q_{n,m}(z)w^m+Q_{n,m-1}(z)w^{m-1}+\dots+Q_{n,0}(z),
\end{equation}
\tag{2}
$$
recover asymptotically $m$ values of the original function $f$. More precisely, these zeros recover the values of $f$ on all the sheets of the Nuttall partition of its Riemann surface, except for the last sheet, outside the compact set $F_m$, which is the projection of the boundary of the last sheet (the definition of the Nuttall partition is given in § 2). At the same time all zeros of the polynomials $Q_{n,j}$, except for a finite number of them, are attracted to the compact set $F_m$ in the limit (as $n\to\infty$); see part (1) of Theorem 1. In this work we study the asymptotic behaviour and the limit zero distribution of the discriminant of the polynomial $P_n(z,w)$ (as a polynomial in $w$), which is denoted by $D_n(z)$. Recall that the discriminant of a polynomial $P(w):=a_N w^N+a_{N-1}w^{N-1}+\dots+ a_0$ of degree $N$ is
$$
\begin{equation}
D(P)=D(a_0,\dots,a_N):=a_N^{2N-2}\prod_{1\leqslant j<k\leqslant N}(r_j-r_k)^2,
\end{equation}
\tag{3}
$$
where $r_1,\dots, r_N$ are the zeros of $P(w)$ written with multiplicities. So the discriminant $D(P)$ vanishes if and only if the polynomial $P$ has a multiple root. It is well known that $D(a_0,\dots,a_N)$ is a homogeneous polynomial of degree $2N-2$ in the variables $a_0,a_1,\dots a_N$. For more information on the properties of the discriminant, see, for example, [32], § 43. In Theorem 2 we find the weak asymptotics of the discriminants $D_n(z)$ and the asymptotics of their ratio with $Q_{n,m}^{2m-2}$. It appears that the weak asymptotics of $D_n$ coincides, up to normalization, with the weak asymptotics of the Hermite–Padé polynomials $Q_{n,j}$ themselves. In particular, all but finitely many zeros of $D_n$ are attracted to the compact set $F_m$ in the limit (as $n\to\infty$). The latter statement, which is part (1) of Theorem 2, was announced in [23], Theorem 1, in the simplest case $m=2$. In Theorem 3 we show that some zeros not attracted by $F_m$ are attracted to some points away from $F_m$, moreover, at an exponential rate. These points of attraction are determined by the function $f$ and they are the zeros in $\mathbb{C}\setminus F_m$ of the function $\Pi_m(z)$ from (12). In Corollary 1 and Theorem 4 we state a sufficient condition for a branch point of the function $f$ not belonging to $\mathbb{C}\setminus F_m$ to attract at least one zero of $D_n$ at an exponential rate. Note that attracting branch points in Theorem 4, unlike Corollary 1, are not always zeros of the function $\Pi_m(z)$. Corollaries 2–4 provide conditions on $f$ that are sufficient in order that each branch point of $f$ not belonging to $\mathbb{C}\setminus F_m$ attracts at least one zero of $D_n$ at an exponential rate. So we show that in these cases the zeros of discriminants $D_n$ can be used to find branch points of $f$ in $\mathbb{C}\setminus F_m$ approximately with exponential accuracy. The question of finding branch points of the original algebraic function $f$ approximately from the coefficients $c_k$ arises in a series of practical problems, for example, in searching for Katz points (see [19]) for different excited states of low-atomic molecules or, more generally, in searching for branch points of the energy functions of low-atomic molecules in various approximations; see [37], [38], [15], [25], [10] and [14]. Note that these works used (without any mathematical justification) a much more general construction than in the model case under consideration here. In problems considered in these works the function $f$ is an algebraic function of a high degree $m+1$, say, about 1000, and it makes no sense to consider the discriminants of polynomials of degree one lower for it. Therefore, in applied problems one considers for such $f$ the Hermite–Padé polynomials $\widetilde Q_{n,j}$ of type I, $j=0, \dots, l$ (where, as a rule, $l=2,3,4$), for the tuple of germs $[1,f_\infty,\dots,f_\infty^l]$, which are defined by the conditions $\deg{\widetilde Q_{n,j}}\leqslant{n}$ and
$$
\begin{equation}
\widetilde Q_{n,0}(z) + \sum_{j=1}^{l}f_\infty^j(z)\widetilde Q_{n, j}(z)=O\biggl(\frac{1}{z^{l(n+1)}}\biggr) \quad \text{as } z\to\infty,
\end{equation}
\tag{4}
$$
and the discriminants $\widetilde D_n$ corresponding to them. Note that if $l=2$, then $\widetilde D_n$ is the usual discriminant of the quadratic equation: $\widetilde D_n=\widetilde Q_{n,1}^2-4\widetilde Q_{n,0}\widetilde Q_{n,2}$. But it is worth noting that in [15] the authors considered zeros of discriminants $\widetilde D_n$ constructed for $l$ up to $l=14$, and in [37], up to $l=20$. In all applied works mentioned it was stated (as noted before, without any mathematical justification) that some zeros of the discriminants $\widetilde D_n$ that are ‘close to the point $z_0$’ at which the Hermite–Padé polynomials are considered (in our situation $z_0=\infty$) converge to some branch points of $f$ ‘close’ to $z_0$ at a ‘very high’ rate. We emphasize that to obtain our model situation one must assume that $f$ is an algebraic function of degree $l+1$, which is not interesting for practical problems. Nevertheless, the case of a general algebraic function is qualitatively covered by the model case if one assumes the validity of Nuttall’s general conjecture, which was put forward in its original form by Nuttall [30] and formulated in other works in one form or another (see, for example, [20], [6], [7] and [34]). In the most general form this conjecture is formulated for an arbitrary algebraic function as follows. Let $f$ be an algebraic function of degree $m+1$. Consider the Hermite–Padé polynomials of type I $\widetilde Q_{n,j}$ for the tuple of germs $[1,f_\infty,\dots,f_\infty^l]$ (4), where $l<m$. Then there exists an $(l+1)$-sheeted branched covering of the Riemann sphere by a compact Riemann surface $\mathfrak{N}_{l+1}$ (the so-called $(l+1)$-sheeted Nuttall surface) such that:
Note that it was proved in [24], Lemma 5, that the complement to the closure of the topmost sheet of the Nuttall partition of any Riemann surface is a domain (that is, it is connected). This justifies part (1) of Nuttall’s conjecture. Part (2) of the conjecture can be refined if one narrows the class of algebraic functions under consideration. Let $f$ be an algebraic function such that all of its branch points are of the second order and it has no pair of branch points over the same point on the Riemann sphere. Then part (2) of Nuttall’s conjecture can be strengthened in the following way. Let $g$ be an arbitrary meromorphic function on $\mathfrak N_{l+1}$ such that the functions $1, g, g^2,\dots, g^{l}$ are independent over the field of rational functions $\mathbb C(z)$. Then the asymptotic behaviour of Hermite–Padé polynomials for the tuple of germs $[1,f_\infty,\dots,f_\infty^l]$ and of objects constructed from these polynomials coincides with the asymptotic behaviour of Hermite–Padé polynomials and the corresponding objects for the tuple of germs $[1,g_\infty,\dots,g_\infty^l]$, where $g_\infty$ is the germ of the function $g$ at the preimage of $\infty$ on the zeroth Nuttall sheet $\mathfrak N_{l+1}^{(0)}$ of the surface $\mathfrak N_{l+1}$. Note that in this refined variant of Nuttall’s conjecture one cannot completely abandon the above condition on the branch points of the function $f$, as wonderfully demonstrated in [48] using numerical simulations. The authors of [48] considered the polynomials $\widetilde Q_{n,0}, \widetilde Q_{n,1}, \widetilde Q_{n,2}$ (so that $l=2$) for $f=\sqrt[4]{(1/z-a_1)(1/z-a_2)(1/z-a_3)(1/z-a_4)}$ for some $a_1,\dots,a_4\in\mathbb{C}$ (so that $m=3$) and showed that the limit zero distributions of the polynomials $\widetilde Q_{n,0}$ and $\widetilde Q_{n,1}$ are different, while, according to Theorem 1, the limit zero distributions of the zeros of Hermite–Padé polynomials for the tuple of germs $[1,g_\infty,g_\infty^2]$, where $g_\infty$ is a germ of an algebraic function of degree 3, must coincide. Nuttall’s conjecture is most often formulated for $l=2$, that is, when the Nuttall surface is three-sheeted (see, for example, [46] and [43]). Although Nuttall’s conjecture has in general not been proved, in all cases when one manages to describe the asymptotic behaviour of the Hermite–Padé polynomials for germs of multivalued functions, this behaviour is expressed just in terms of such a Nuttall surface. As a rule, in these cases the functions have some reality properties, and in most cases the Nuttall surface is constructed using the solution of a certain vector potential-theory equilibrium problem which appeared originally in [16]; see, for example, [2], [17], [5], [3], [33], [4], [27], [28] and [34]. Note that sometimes a scalar potential-theory equilibrium problem is used for constructing the Nuttall surface; see, for example, [35], [20], [47], [44] and [49]. Suetin in [45] and in his joint work with Dobrolyubov, Ikonomov and Knizhnerman [13] managed to formalize the statements of the authors of the applied works mentioned above. In [45] the discriminants $\widetilde D_n=\widetilde Q_{n,1}^2-4\widetilde Q_{n,0}\widetilde Q_{n,2}$ were considered, where $\widetilde Q_{n,0}$, $\widetilde Q_{n,1}$ and $\widetilde Q_{n,2}$ are the Hermite–Padé polynomials of type I constructed from the tuple of germs $[1,f_\infty,f_\infty^2]$ (that is, the simplest case when $l=2$ in (4) was considered), where $f_\infty$ is the germ of a not even algebraic but multivalued analytic function $f$, although with very strong restrictions on the nature of branch points. In [45] the following conjecture was put forward: each branch point of such a function $f$ that belongs to the Stahl compact set for the germ $f_\infty$ (see below on the Stahl compact set in more detail) attracts at least one zero of the discriminant $\widetilde D_n$. Corollary 4 in our paper covers in fact this conjecture in the model case when $f$ is an algebraic function of degree 3, and moreover, imposes no additional conditions on the nature of branch points or the Stahl compact set, since, in accordance with part (3) of Theorem 2, the zeros of the discriminants $D_n$ fully model the compact set $F_2$. In [13], as also in [45], the discriminants $\widetilde D_n=\widetilde Q_{n,1}^2-4\widetilde Q_{n,0}\widetilde Q_{n,2}$ (that is, when ${l=2}$ in (4)) were considered. The authors of [13] suggested (without a mathematical justification) and tested in practice searching for some branch points of an algebraic function with the help of such discriminants $\widetilde D_n$. In doing so, they put forward implicitly a conjecture that we formulate below. The general Nuttall conjecture states that the asymptotic behaviour of such Hermite–Padé polynomials $\widetilde Q_{n,j}$ and discriminants $\widetilde D_n$ must be determined by a three-sheeted Nuttall surface $\mathfrak{N}_3$. At the same time, in accordance with Nuttall’s conjecture and Theorems 1 and 2, for a generic algebraic function almost all the zeros of the Hermite–Padé polynomials and discriminants tend in the limit to the compact set $F_2$, that is, the projection of the boundary of the topmost Nuttall sheet $\mathfrak{N}^{(2)}_3$ of the surface $\mathfrak{N}_3$. Moreover, it is known that in the model case when the degree of the function $f$ equals 3 (and therefore, supposing that Nuttall’s conjecture holds true, also for a generic algebraic function), almost all zeros of the Hermite–Padé polynomials of type II tend to the compact set $F_1$, that is, the projection of the boundary of the bottom Nuttall sheet $\mathfrak{N}^{(0)}_3$ of the surface $\mathfrak{N}_3$; see [30] and also [22]. It is known (see [24], Lemma 3) that the compact sets $F_1$ and $F_2$ are the closures of finite unions of analytic arcs which are $C^{1+\alpha}$-smooth at their endpoints for some $\alpha>0$ (that is, the intersection of such an arc with a sufficiently small neighbourhood of an endpoint is the image of the half-open interval $[0,1)$ under a $C^{1+\alpha}$-embedding in this neighbourhood). It is not difficult to show that in the model case any endpoint of the compact set $F_1$ or $F_2$ (that is, a point from which a unique analytic arc goes out) is a branch point of the original algebraic function $f$. The authors of [13] put forward, in fact, the conjecture that at least one zero of the discriminant $\widetilde D_n$ tends to any endpoint of $F_1$ that does not belong to $F_2$. Moreover, it follows from their numerical calculations that the convergence of such a zero is very rapid, at least of exponential type. Conjecture 4 here does not only justify this conjecture in the model case (for an algebraic function $f$ of degree 3) but also proves a stronger result in this model case: any branch point of $f$ in $\mathbb{C}\setminus F_2$ attracts at least one zero of the discriminant at an at least exponential rate. Let us discuss the exponential rate of convergence of a zero of $D_n$ to a branch point of $f$ that is obtained in Theorem 4 and Corollaries 1–4, and let us compare it with the rate of identification of branch points $f$ in other approaches. The simplest way to find some branch points of a general algebraic function $f$ is to use classical Padé polynomials (to define them one sets $l=1$ in (4)). In contrast to Hermite–Padé polynomials, for Padé polynomials constructed from the germ of an algebraic function we have Stahl’s general theorem (see [39]–[42], and also the survey article [1]), which describes their weak asymptotic behaviour. Moreover, in [8] their strong asymptotics was found in the case of a generic algebraic function. According to Stahl’s theorem, all the zeros of Padé polynomials, except for $o(n)$ of them, tend to the so-called Stahl compact set $S$, and their normalized counting measure converges to the equilibrium measure $\lambda_S$ of $S$. It is well known (see [31], [9] and [8]) that the Stahl compact set $S$ is the closure of a finite union of analytic arcs, and it is easy to show that at each endpoint $x_0$ of $S$ the equilibrium measure $\lambda_S$ shows the behaviour $O(|z-x_0|^{-1/2})\,|dz|$. It easily follows from Stahl’s general theory that all the endpoints of $S$ are branch points of the function $f$. Thus, Padé polynomials can be used to find these branch points. However, in view of the behaviour of the equilibrium measure $\lambda_S$ mentioned above, one can only guarantee a quadratic rate of convergence of a zero of Padé polynomials to the point $x_0$, an endpoint of $S$, that is, one can assert that the $n$th Padé polynomial has a zero at a distance of at most $O(1/n^2)$ from $x_0$. If we return to our model situation (1) of an algebraic function $f$ of degree $m+1$, then it follows from Theorem 1 that the zeros of the Hermite–Padé polynomials of type I tend to the compact set $F_m$. Again, it is easy to see that endpoints of $F_m$ are branch points of $f$, and therefore to find them approximately, we can use Hermite–Padé polynomials of type I. However, we can show that only a polynomial rate of convergence of the zero of the polynomial $Q_{n,j}$ to any endpoint of $F_m$ follows from part (2) of Theorem 1. Moreover, in [22] (see also [21]) a polynomial Hermite–Padé $m$-system was introduced, which includes the Hermite–Padé polynomials of types I and II. In [22] it was shown that for a generic algebraic function of degree $m+1$ the zeros of the $k$th polynomials of the Hermite–Padé $m$-system, $k=1,\dots,m$, tend to certain compact sets $F_k$ (a rigorous definition of the $F_k$ is given in § 2, in the description of the Nuttall partition, and in the generic case one can say that $F_k$ is the projection of the boundary between the $(k-1)$st and $k$th sheets of the Nuttall partition of the Riemann surface of $f$). Again, it is easy to see that all the endpoints of $F_k$ are branch points of $f$. Thus, the zeros of the $k$th polynomials of the Hermite–Padé $m$-system can be used to find such branch points approximately. However, in this case it also follows from part 3 of Theorem 1 in [22] that we can only guarantee a polynomial rate of convergence of the zero of the $k$th polynomial of the Hermite–Padé $m$-system to such a branch point. Corollary 3 shows that in our model situation, given a generic algebraic function, we can guarantee an exponential (geometric) rate of convergence of a zero of the discriminant $D_n$ to any branch point of $f$ in $\mathbb{C}\setminus F_m$. Thus, zeros of the discriminants $D_n$ provide a fundamentally faster method for finding branch points of the function $f$ numerically than methods based on the use of zeros of rational approximants. Taking Nuttall’s general conjecture into account, it can be expected that in the general case zeros of the discriminants $\widetilde D_n$ also provide a method for finding branch points of the original function at an exponential rate. The paper is organized as follows. Section 2 describes in detail the construction of the Nuttall partition of a Riemann surface into sheets, provides for completeness the formulation of Theorem 1, proved in [24], formulates Theorems 2–4 proved in this paper, and deduces Corollaries 1–4 from Theorems 3 and 4. In § 3 we introduce the necessary notions and prove the technical results of Proposition 1 and Lemma 1, which are used later to prove Theorems 3 and 4. Section 4 is devoted to the proof of Theorem 2, and § 5 is devoted to the proofs of Theorems 3 and 4. § 2. The Nuttall partition and the formulation of the main resultsWe repeat once again that in this paper we study only the model situation, when we consider the Hermite–Padé polynomials $Q_{n,j}$ of type I for the tuple of germs $[1, f_\infty, f_\infty^2,\dots, f_\infty^m]$ (1) and the discriminants $D_n$ of the polynomials (2), where $f_\infty$ is the germ of an algebraic function $f$ of degree precisely $m+1$. Theorems 1–4 are formulated and proved only for this case. Let us introduce the necessary notation related to the Riemann surface of $f$. Throughout what follows $f(z)$ is the solution of the equation $\mathcal{P}(z,f(z))\equiv 0$, where $\mathcal{P}(z,w)$ is an irreducible polynomial in $z$ and $w$ and $\deg_w \mathcal{P}(z,w)=m+1$. Let $\mathfrak{R}$ be the standard compactification of the Riemann surface of $f$, which is obtained from the closure of the graph $\{(z,w)\colon \mathcal{P}(z,w)=0\}$ in $\widehat{\mathbb{C}}_z\times\widehat{\mathbb{C}}_w$ by the standard procedure of resolution of singularities (see, for example, [26], § 13.2). Then a natural projection is defined on $\mathfrak{R}$, acting as $(z,w)\mapsto z$ on a nonsingular points of the graph, which realizes an $(m+1)$-sheeted branched holomorphic covering of $\widehat{\mathbb{C}}$ by the surface $\mathfrak{R}$. In order not to confuse this projection with the $z$-coordinate we denote it by $\pi$, that is, $\pi\colon (z,w)\mapsto z$ at nonsingular points of the graph. For points on $\mathfrak{R}$ we use bold symbols, for example $\mathbf{z}$, and their projections are denoted by the corresponding ‘ordinary’ characters, for example, $z$, that is, $z=\pi(\mathbf{z})$. The function $f$ lifts naturally to a meromorphic function $f(\mathbf{z})$ on $\mathfrak{R}$. In this case the given holomorphic germ $f_\infty(z)$ lifts to the germ of $f(\mathbf{z})$ at a distinguished point in the set $\pi^{-1}(\infty)$, which we denote by $ \infty^{(0)}$. We emphasize that we allow the case when $\infty$ is a branch point of $f(z)$, but the point $ \infty^{(0)}$ is not a critical point of $\pi$. We can also define the objects we consider (the function and its germ) in a more abstract way, without being tied to a specific algebraic equation. We can assume that an $(m+1)$-sheeted holomorphic branched covering $\pi:\mathfrak{R}\to\widehat{\mathbb{C}}$ of the Riemann sphere $\widehat{\mathbb{C}}$ by an abstract compact Riemann surface $\mathfrak{R}$ is fixed and a point $ \infty^{(0)}\in\pi^{-1}(\infty)$ is distinguished, which is not a critical point of the projection $\pi$. We consider an arbitrary meromorphic function $f(\mathbf{z})$ on $\mathfrak{R}$ such that the functions $1, f, f^2, \dots, f^m$ are independent over the field of rational functions $\mathbb{C}(z)$. Then the germ $f_\infty(z)$, from which we construct the Hermite–Padé polynomials (1), is the germ of $f(\mathbf{z})$ at the point $ \infty^{(0)}$, considered as a function of $z$, that is, $f_\infty(z)=f(\pi_0^{-1}(z))$, where $\pi^{-1}_0$ is the inverse mapping of $\pi$ in a neighbourhood of the point $ \infty^{(0)}$. Note that any function $\varphi(z)$ in a domain $D\subset\widehat{\mathbb{C}}$ lifts naturally to $\pi^{-1}(D)\subset\mathfrak{R}$ as $\varphi(\mathbf{z}):=\varphi\circ\pi(\mathbf{z})=\varphi(z)$. Without complicating the notation, for such lifts of $\varphi(\mathbf{z})$ we use the same symbol $\varphi(z)$ as for the original function, specifying its domain of definition if necessary. Let us define a key concept for us, namely, the Nuttall partition of the Riemann surface $\mathfrak{R}$ into sheets, introduced by Nuttall [29], [30]. In fact, the Nuttall partition is constructed for a finite-sheeted holomorphic branched covering of the Riemann sphere $\widehat{\mathbb{C}}$ by a compact Riemann surface with a distinguished point that is not critical for the projection. However, it has historically turned out that one speaks about the Nuttall partition of a Riemann surface, implying implicitly that a natural projection of it (for example, as of the Riemann surface of an algebraic function) onto the Riemann sphere and a distinguished point on it, at which the holomorphic germ is considered, are fixed. In our case the Riemann surface is $\mathfrak{R}$, the projection is $\pi$, and the distinguished point is $ \infty^{(0)}$. In order to keep the notation simple and follow tradition, we will also speak about the Nuttall partition of the surface $\mathfrak{R}$. In the definition of the Nuttall partition we follow [24], where all the requisite properties of this partition were proved. So, let $u(\mathbf{z})$ be a harmonic function in $\mathfrak{R}\setminus\pi^{-1}(\infty)$ with the following logarithmic singularities at points in $\pi^{-1}(\infty)$:
$$
\begin{equation}
\begin{aligned} \, u(\mathbf z)&=-m\log{|z|}+O(1), \qquad\mathbf z\to \infty^{(0)}, \\ u(\mathbf z)&=\log{|z|}+O(1), \qquad\mathbf z\to\pi^{-1}(\infty)\setminus \infty^{(0)}. \end{aligned}
\end{equation}
\tag{5}
$$
The function $u$ always exists and is defined up to an additive constant (it is constructed explicitly using standard bipolar Green’s functions; see [24], formula (23)). Let $z\in\mathbb{C}$, and let $u_0(z),\dots,u_m(z)$ be the values of the function $u$ at the points of the set $\pi^{-1}(z)$, ordered in nondecreasing order (and in the case when $z$ is the critical value of $\pi$, repeated in accordance with the order of the corresponding point of the set $\pi^{-1}(z)$ as a critical point of $\pi$):
$$
\begin{equation}
u_0(z)\leqslant u_1(z)\leqslant\dots\leqslant u_{m-1}(z)\leqslant u_m(z).
\end{equation}
\tag{6}
$$
If $u_{j-1}(z)< u_j(z)<u_{j+1}(z)$, then in the set $\mathfrak{R}^{(j)}$ ($j$th sheet of the surface $\mathfrak{R}$, $j=0,\dots,m$) we include the point $\mathbf{z}^{(j)}\in\pi^{-1}(z)$ such that $u(\mathbf{z}^{(j)})=u_j(z)$ (for ${j=0}$ we consider only the inequality $u_0(z)<u_{1}(z)$, and for $j=m$ only the inequality $u_{m-1}(z)<u_{m}(z)$). Otherwise, the points of the set $\pi^{-1}(z)$ are not included in $\mathfrak{R}^{(j)}$. For $z=\infty$, $u(z)$ must be replaced by $u(z)-\log |z|$ in (6). Thus, formally, the sheets $\mathfrak{R}^{(j)}$ are defined by
$$
\begin{equation*}
\begin{aligned} \, \mathfrak R^{(0)} &:=\{\mathbf z\in\mathfrak R\colon 0<u_{1}(z)-u(\mathbf z)\}; \\ \mathfrak R^{(j)} &:=\{\mathbf z\in\mathfrak R\colon u_{j-1}(z)-u(\mathbf z)<0<u_{j+1}(z)-u(\mathbf z)\}, \qquad j=1,\dots,m-1; \\ \mathfrak R^{(m)} &:=\{\mathbf z\in\mathfrak R\colon u_{m-1}(z)-u(\mathbf z)<0\}. \end{aligned}
\end{equation*}
\notag
$$
It follows from the definition that the $\mathfrak{R}^{(j)}$ are disjoint open (in general, disconnected) subsets of $\mathfrak{R}$ and the projection $\pi\colon \mathfrak{R}^{(j)}\to\pi(\mathfrak{R}^{(j)})$ is biholomorphic. Throughout the paper, a point in $\mathfrak{R}^{(j)}$ lying over $z\in \widehat{\mathbb{C}}$ is denoted by $\mathbf{z}^{(j)}$ and the boundary of the sheet $\mathfrak{R}^{(j)}$ by $\partial\mathfrak{R}^{(j)}$. The point $ \infty^{(0)}$ distinguished before always belongs to the sheet $\mathfrak{R}^{(0)}$, which agrees with our notation for points on sheets. It is clear that no critical point of the projection $\pi$ lies in any set $\mathfrak{R}^{(j)}$. Let
$$
\begin{equation}
\begin{gathered} \, F_j:=\{z\in\widehat{\mathbb C}\colon u_{j-1}(z)=u_j(z)\}, \quad j=1,\dots,m, \\ F:=\bigcup_{j=1}^mF_j. \end{gathered}
\end{equation}
\tag{7}
$$
It was shown in [24], Appendix 1, that all sets $F_j$ and $\partial\mathfrak{R}^{(j)}$ are (real) one-dimensional piecewise analytic sets without isolated points. The precise definition of a piecewise analytic set is presented there. Informally speaking, this means that these sets are the closures of unions of a finite number of analytic arcs that have some regularity at their endpoints. In particular, it follows from this that the sets $F_j$ have no interior, and this immediately implies that $\pi(\partial\mathfrak{R}^{(j)})=F_j\cup F_{j+1}$ for $j=1,\dots,m-1$, as well as $\pi(\partial\mathfrak{R}^{(0)})=F_1$ and $\pi(\partial\mathfrak{R}^{(m)})=F_m$. Let us introduce some notation used in what follows. We denote by $\xrightarrow{*}$ weak $*$-convergence, indicating the space in which it is considered if necessary, and by $\xrightarrow{\mathrm{cap}}$ convergence in (logarithmic) capacity, specifying the set on which it is considered. We denote the distance in the spherical metric on $\widehat{\mathbb{C}}$ by $\operatorname{dist}$. Let
$$
\begin{equation*}
d\sigma:= \frac{i}{2\pi}\frac{dz\wedge d\mkern2mu\overline z}{(1+|z|^2)^2}
\end{equation*}
\notag
$$
be the normalized area form of the spherical metric on $\widehat{\mathbb{C}}$. Recall that the operator $\operatorname{dd^c}$ is the standard analogue of the Laplace operator on Riemann surfaces, which, in general, maps currents of degree 0 there to currents of degree 2 and acts on smooth functions $\varphi$ as $\operatorname{dd^c}\varphi = (\varphi_{xx}+\varphi_{yy})\,dx\,dy=\Delta\varphi\,dx\,dy$ in the local coordinate $\zeta = x +iy$. All the properties of the operator $\operatorname{dd^c}$ and the statements from potential theory on compact Riemann surfaces that we need are presented in [24], Appendix 2, and [11]. We will refer to them, preserving the notation whenever possible. To discuss the asymptotic behaviour of the Hermite–Padé polynomials $Q_{n,j}$ and the discriminants $D_n$, it is necessary to fix their normalizations. Therefore, along with the polynomials $Q_{n,j}$ defined by (1) and the discriminants $D_n$ of the equations (2), we will consider the polynomials $Q_{n,j}^*:=c_{n,j}Q_{n,j}$ and $D_n^*(z):=d_nD_n(z)$ ($c_{n,j}>0$ and $d_n>0$ are constants) such that the functions $\log|Q_{n,j}^*|$ and $\log|D_n^*|$ are spherically normalized:
$$
\begin{equation}
\int_{\widehat{\mathbb C}}\log|Q_{n,j}^*|\,d\sigma=0 \quad\text{and}\quad \int_{\widehat{\mathbb C}}\log|D_n^*|\,d\sigma=0.
\end{equation}
\tag{8}
$$
Note that, in general, $Q_{n,j}^*$ do not satisfy (1). For the function $u(\mathbf{z})$ we choose the spherical normalization on the $m$th sheet of $\mathfrak{R}^{(m)}$, that is, we normalize it by the condition
$$
\begin{equation}
\int_{\widehat{\mathbb C}\setminus F_m} u(\mathbf z^{(m)})\, d\sigma(z)=0.
\end{equation}
\tag{9}
$$
Then $u_m(z)$ is spherically normalized: Let $\sigma_j$, $j=1,\dots,m$, be the $j$th elementary symmetric polynomial in $m$ variables (the sum of all possible products of $j$ distinct variables of $m$). Set
$$
\begin{equation}
\Xi_j(z):=(-1)^{m-j}\sigma_{m-j}\bigl(f(\mathbf z^{(0)}), f(\mathbf z^{(1)}),\dots, f(\mathbf z^{(m-1)})\bigr).
\end{equation}
\tag{11}
$$
The functions $\Xi_j(z)$ are well defined in $\widehat{\mathbb{C}}\setminus F$ (where all the $\mathbf{z}^{(j)}$ are defined) by formula (11) and meromorphic there. However, since $\sigma_j$ is a symmetric function, we see that $\Xi_j(z)$ depend only on the values of $f$ at the points of $\pi^{-1}(z)\setminus \mathbf{z}^{(m)}$ but not on their order. Thus, $\Xi_j(z)$ can be extended to meromorphic functions in $\widehat{\mathbb{C}}\setminus F_m$. (For more details, see a similar statement for $\Pi_m$ in Remark 1.) The following theorem (Theorem 1) was proved in [24]. More precisely, Theorem 1 is the combination of Theorems 1 and 2 in [24] in the special case when $f_j=f^j$, where $f$ is a meromorphic function on $\mathfrak{R}$. We present its formulation for completeness. It should be noted that in [24], instead of part (2) of Theorem 1, only weak convergence in all spaces $L^p$ for $p>1$ was proved, and the assertion of part (2) of Theorem 1 presented here was proved in [22], Theorem 1, as the case $k=m$ there. Theorem 1. (1) There exists an integer $L\in\mathbb N$ such that for any neighbourhood $V$ of the compact set $F_m$, for all sufficiently large $n$, $n>N(V)$, at most $L$ zeros of each polynomial $Q_{n,j}$, $j=0,\dots,m$, lie outside the neighbourhood $V$. (2) For any $p\in[1;\infty)$, as $n\to\infty$,
$$
\begin{equation*}
\frac{1}{n}\log|Q_{n,j}^*(z)|\to u_m(z) \quad \textit{in } L^p(\widehat{\mathbb C}, d\sigma).
\end{equation*}
\notag
$$
(3) As $n\to\infty$,
$$
\begin{equation*}
\frac 1 n \operatorname{dd^c}\log|Q_{n,j}|\xrightarrow{*}\operatorname{dd^c} u_m.
\end{equation*}
\notag
$$
(4) For any compact set $K\subset\mathbb{C}\setminus F_m$, as $n\to\infty$,
$$
\begin{equation*}
\frac {Q_{n,j}(z)}{Q_{n,m}(z)}\xrightarrow{\operatorname{cap}}\Xi_j(z), \qquad z\in K.
\end{equation*}
\notag
$$
(5) Moreover, for any $\varepsilon>0$ Let
$$
\begin{equation}
\Pi_m(z):=\prod_{0\leqslant j<k\leqslant m-1}(f(\mathbf z^{(j)})-f(\mathbf z^{(k)}))^2.
\end{equation}
\tag{12}
$$
Just like the functions $\Xi_j$ in (11), the function $\Pi_m(z)$ is well defined by the formula (12) only in $\widehat{\mathbb{C}}\setminus F$, but it can be extended to a meromorphic function in $\widehat{\mathbb{C}}\setminus F_m$. The proof of the following theorems (Theorems 2–4) form the main content of this paper. Theorem 2. (1) There exists an integer $L'\in\mathbb N$ such that for any neighbourhood $V$ of the compact set $F_m$, for all sufficiently large $n$, $n>N(V)$, at most $L'$ zeros of each polynomial $D_n$ lie outside the neighbourhood $V$. (2) For any $p\in[1;\infty)$, as $n\to\infty$,
$$
\begin{equation}
\frac{1}{(2m-2)n}\log|D_{n}^*(z)|\to u_m(z) \quad \textit{in } L^p(\widehat{\mathbb C}, d\sigma).
\end{equation}
\tag{13}
$$
(3) As $n\to\infty$,
$$
\begin{equation}
\frac {1}{(2m-2)n} \operatorname{dd^c}\log|D_n|\xrightarrow{*}\operatorname{dd^c} u_m.
\end{equation}
\tag{14}
$$
(4) For any compact set $K\subset\mathbb{C}\setminus F_m$, as $n\to\infty$,
$$
\begin{equation*}
\frac {D_n(z)}{Q_{n,m}^{2m-2}(z)}\xrightarrow{\operatorname{cap}}\Pi_m(z), \qquad z\in K.
\end{equation*}
\notag
$$
(5) Moreover, for any $\varepsilon>0$ Theorem 3. Let $a\in\mathbb{C}\setminus F_m$ be a point such that $\Pi_m(a)=0$. Then there exist constants $c>0$ and $0<A<1$ such that for any sufficiently large $n\in\mathbb{N}$, $n>N$, there exists a point $z_{n}\in\mathbb{C}$ with the properties It is clear that if $a\in\mathbb{C}\setminus F_m$ is a branch point of $f$ such that the function $f$ has no poles at points in $\pi^{-1}(a)$, then $\Pi_m(a)=0$. Therefore, Theorem 3 clearly implies the following. Corollary 1. Let $a\in\mathbb{C}\setminus F_m$ be a branch point of $f$ (a critical value of $\pi$) such that $f$ has no poles in the set $\pi^{-1}(a)$. Then there exist constants $c>0$ and $0<A<1$ such that for any sufficiently large $n\in\mathbb{N}$, $n>N$, there exists a point $z_{n}\in\mathbb{C}$ with the properties A sufficient condition for an algebraic function $f(z)$ not to have poles in $\mathbb{C}$ at all is clearly the monicity of the polynomial $\mathcal{P}(z,w)$ defining $f$ (as a polynomial in $w$), so that $\mathcal{P}(z,w)=w^{m+1}+\dotsb$. Therefore, Corollary 1 implies another result. Corollary 2. Let the algebraic function $f(z)$ be the solution of the polynomial equation $\mathcal{P}(z,f(z))=0$, where $\mathcal{P}(z,w)$ is an irreducible polynomial, $\deg\mathcal{P}_w=m+1$ and $\mathcal{P}(z,w)=w^{m+1}+\dotsb$. Then for any branch point $a$ of $f$ belonging to $\mathbb{C}\setminus F_m$ there exist constants $c>0$ and $0<A<1$ such that for any sufficiently large $n\in\mathbb{N}$, $n>N$, there exists a point $z_{n}\in\mathbb{C}$ with the properties Theorem 4. Let $a\in\mathbb{C}\setminus F_m$ be a branch point of $f$ (a critical value of $\pi$), and there is only one critical point of $\pi$ in the set $\pi^{-1}(a)$, which is of the second order. Then there exist constants $c>0$ and $0<A<1$ such that for any sufficiently large $n\in\mathbb{N}$, $n>N$, there exists a point $z_{n}\in\mathbb{C}$ with the properties Theorem 4 obviously implies the following result. Corollary 3. Let the projection $\pi\colon \mathfrak{R}\to\widehat{\mathbb{C}}$ be such that all of its critical points are of the second order and there is at most one critical point of $\pi$ over each point $z\in\widehat{\mathbb{C}}$. Then for any branch point $a$ of $f$ (a critical value of $\pi$) belonging to $\mathbb{C}\setminus F_m$ there exist constants $c>0$ and $0<A<1$ such that for any sufficiently large $n\in\mathbb{N}$, $n>N$, there exists a point $z_{n}\in\mathbb{C}$ with the properties Consider the case when $m=2$. In this case $f$ is an algebraic function of order $3$, and the discriminants $D_n=Q_{n,1}^2-4Q_{n,0}Q_{n,2}$ are usual discriminants of a quadratic equation. So the projection $\pi$ (also denoted by $z$) realizes a three-sheeted covering of the Riemann sphere $\widehat{\mathbb{C}}$ by the Riemann surface $\mathfrak{R}$ of $f$. Then there can clearly be only one critical point of $\pi$ over each point $a\in\widehat{\mathbb{C}}$, and its order must be $2$ or $3$. If the order of such a critical point is $3$, then for the functions $u_j$ from (6) we have $u_0(a)=u_1(a)=u_2(a)$. Therefore, in this case, from the definition of (7) we obtain $a\in F_m=F_2$. So for $m=2$ any branch point $a$ of the function $f$ lying outside $F_2$ satisfies the assumptions of Theorem 4, and we obtain another corollary of this theorem. Corollary 4. Let $m=2$. Then for any branch point $a$ of the function $f$ belonging to $\mathbb{C}\setminus F_2$ there exist constants $c>0$ and $0<A<1$ such that for any sufficiently large $n\in\mathbb{N}$, $n>N$, there exists a point $z_{n}\in\mathbb{C}$ with the properties § 3. Preparatory resultsWe define the remainder function $R_n(\mathbf{z})$ on $\mathfrak{R}$ by the formula
$$
\begin{equation*}
R_n(\mathbf z):=Q_{n,0}(z) + \sum_{j=1}^{m}f^j(\mathbf z)Q_{n, j}(z).
\end{equation*}
\notag
$$
Clearly, $R_n(\mathbf{z})$ is meromorphic on $\mathfrak{R}$. The defining relation for the Hermite–Padé polynomials (1) means that $R_n(\mathbf{z})= O(z^{-m(n+1)})$ as $\mathbf{z}\to \infty^{(0)}$. The key role in the proof of Theorem 1 given in [24] is played by the fact that the Hermite–Padé polynomials $Q_{n,j}$ are expressed in terms of $R_n(\mathbf{z})$ (see (22) below), and the absolute value of the properly normalized function $R_n$ behaves, informally speaking, like $e^{nu(\mathbf{z})}$ (see formula (19) below). We will also need such a representation for $Q_{n,j}$, so we discuss it in more detail. Recall that for any two distinct points $\mathbf{q}, \mathbf{p}\in\mathfrak{R}$ the bipolar Green’s function $g(\mathbf{q}, \mathbf{p}; \mathbf{z})$ with poles $\mathbf{q}$ and $\mathbf{p}$ is a harmonic function on $\mathfrak{R}\setminus\{\mathbf{q}, \mathbf{p}\}$ with logarithmic singularities at $\mathbf{q}$ and $\mathbf{p}$ that have the following form in local coordinates $\zeta$:
$$
\begin{equation}
\begin{aligned} \, g(\mathbf q,\mathbf p;\mathbf z)&=\log{|\zeta(\mathbf z)-\zeta(\mathbf q)|}+O(1), \qquad \mathbf z\to\mathbf q, \\ g(\mathbf q,\mathbf p;\mathbf z)&=-\log|\zeta(\mathbf z)-\zeta(\mathbf p)|+O(1), \qquad \mathbf z\to\mathbf p. \end{aligned}
\end{equation}
\tag{15}
$$
For $\mathbf{q}=\mathbf{p}$ we have $g(\mathbf{p},\mathbf{p};\mathbf{z}) \equiv \mathrm{const}$. The existence of bipolar Green’s functions on any compact Riemann surface (which is well known) and all of their properties we need were proved in [24] (see also [11]). The functions $g(\mathbf{q},\mathbf{p};\mathbf{z})$ are defined up to an additive constant. We assume that they are spherically normalized on the $m$th sheet, that is,
$$
\begin{equation}
\int_{\widehat{\mathbb C}\setminus F_m} g(\mathbf q,\mathbf p;\mathbf z^{(m)})\, d\sigma(z)=0.
\end{equation}
\tag{16}
$$
In particular, $g(\mathbf{p},\mathbf{p};\mathbf{z})\equiv 0$. Let $M_0$ be the sum of the multiplicities of all poles of the function $f(\mathbf{z})$ and $M=mM_0$. It is easy to obtain the following expression for the divisor $(R_n)$ of the remainder function $R_n$ (see [24], formula (17) and Remark 1):
$$
\begin{equation}
(R_n)=n\biggl(m\, \infty^{(0)}-\sum_{l=1}^m \infty_l\biggr)-\sum_{k=1}^{M}\mathbf p_k +\sum_{k=1}^{M}\mathbf q_k(n),
\end{equation}
\tag{17}
$$
where $\{ \infty_l\}_{l=1}^m$ are all points in the set $\pi^{-1}(\infty)\setminus \infty^{(0)}$, and if $ \infty_l$ is a critical point of $\pi$, then it is written as many times as its order; the points $\mathbf{p}_k$, $k=1,\dots, M$, are the poles of the function $f(\mathbf{z})$, each of which is taken with multiplicity $m$ times greater than the order of the pole of $f$ at this point, and the points $\mathbf{q}_k=\mathbf{q}_k(n)$, $k=1,\dots, M$, are the so-called free zeros of the function $R_n(\mathbf{z})$, depending on $n$. Note that, in fact, not all the $\mathbf{q}_k(n)$ depend on $n$: for example, at the point $ \infty^{(0)}$ the function $R_n$ has a zero of order $m(n+1)$, and therefore $ \infty^{(0)}$ is always among the points $\mathbf{q}_k(n)$. Nevertheless, we call all such zeros of $R_n$ free and assume a priori that they depend on $n$. For our further reasoning, it is only important to know about the points $\mathbf{p}_k$ (that is, possible poles of the function $R_n$) that their positions are fixed (do not depend on $n$). From (17) it follows (see [24], Proposition 1) that $|R_n(\mathbf{z})|=C_n e^{nu(\mathbf{z})}\psi_n(\mathbf{z})$, where $C_n>0$ is a constant and
$$
\begin{equation}
\psi_n(\mathbf z):=\exp\biggl\{\sum_{k=1}^M g(\mathbf q_k(n),\mathbf p_k;\mathbf z)\biggr\}.
\end{equation}
\tag{18}
$$
Dividing, if necessary, all Hermite–Padé polynomials $Q_{n,j}$ by $C_n$, we assume that $C_n=1$, that is, this means that the function $\log|R_n|$ is spherically normalized on the $m$th sheet: $\displaystyle \int_{\widehat{\mathbb C}\setminus F_m} \log |R_n(\mathbf z^{(m)})|\, d\sigma(z)=0$ (recall that the function $u(\mathbf{z})$ is also spherically normalized on the $m$th sheet (9)). Throughout the rest of the paper we assume that for each $n$ the normalization of the polynomials $Q_{n,j}$ is chosen so that (19) is satisfied. Following [24], to express the Hermite–Padé polynomials $Q_{n,j}$ in terms of the remainder function $R_n$, we define the matrix $A$ on the set $\widehat{\mathbb{C}}\setminus F$:
$$
\begin{equation}
A(z):= \begin{pmatrix} 1 & f(\mathbf z^{(0)}) & \ldots & f^m(\mathbf z^{(0)})\\ 1 & f(\mathbf z^{(1)}) & \ldots & f^m(\mathbf z^{(1)})\\ \dots & \dots & \dots & \dots \\ 1 & f(\mathbf z^{(m)}) & \ldots & f^m(\mathbf z^{(m)}) \end{pmatrix}
\end{equation}
\tag{20}
$$
(we number the rows and columns of $A$ from $0$ to $m$). Let $A_{k,j}(z)$, $k,j=0,\dots, m$, be the algebraic complement of the element of $A(z)$ with index $(k,j)$. For each $j=0,\dots, m$, on the set $\pi^{-1}(\widehat{\mathbb{C}}\setminus F)$ we define the function In [29] (see also [24], Proposition 2) it was shown that the functions $A_j(\mathbf{z})$ can be extended to meromorphic functions on the whole surface $\mathfrak{R}$. As shown in [24], formula (38), Hermite–Padé polynomials are expressed in terms of the functions $R_n(\mathbf{z})$ and $A_j(\mathbf{z})$ by the following formula, which is valid for all ${z\in\widehat{\mathbb{C}}}$:
$$
\begin{equation}
Q_{n,j}(z)=\sum_{\mathbf z\in\pi^{-1}(z)} R_n(\mathbf z)A_j(\mathbf z),\qquad j=0,\dots,m;
\end{equation}
\tag{22}
$$
in the case when $\mathbf{z}$ is a critical point of the projection $\pi$, the corresponding term should be repeated as many times as its order. At $z\in\widehat{\mathbb{C}}\setminus F_m$ formula (22) can be rewritten as
$$
\begin{equation}
Q_{n,j}(z)=R_n(\mathbf z^{(m)})A_j(\mathbf z^{(m)})(1+h_{n,j}(z)),
\end{equation}
\tag{23}
$$
where
$$
\begin{equation}
h_{n,j}(z)=\sum_{\mathbf z\in\pi^{-1}(z)\setminus\mathbf z^{(m)}} \frac{R_n(\mathbf z)}{R_n(\mathbf z^{(m)})}\frac{A_j(\mathbf z)}{A_j(\mathbf z^{(m)})}, \qquad j=0,\dots,m.
\end{equation}
\tag{24}
$$
The representation (23) is important for the proof of Theorem 1 given in [24]. Now we obtain a similar representation for the discriminants $D_n$, which will equally be important for the proof of Theorems 2–4. Recall that the function $\Pi_m(z)$ was defined by (12). Proposition 1. The following representation of the discriminant $D_n(z)$ is valid in ${\widehat{\mathbb{C}}\setminus F_m}$: where and the $M_{j_1,j_2,\dots,j_m}(z)$ are polynomials with integer coefficients of degree at most $2m-2$ in the functions $A_j(\mathbf{z}^{(m)})$ . Proof. According to our notation (3), Since the discriminant of a polynomial of degree $m$ is a homogeneous polynomial of degree $2m-2$ in its coefficients, substituting the expressions (23) into (27) for the $Q_{n,j}$, we obtain
$$
\begin{equation}
D_n(z)=R_n^{2m-2}(\mathbf z^{(m)})D\bigl(A_0(\mathbf z^{(m)})(1+h_{n,0}(z)), \dots,A_m(\mathbf z^{(m)})(1+h_{n,m}(z))\bigr).
\end{equation}
\tag{28}
$$
The discriminant in the right-hand side of (28) is a polynomial of degree $2m-2$ in the functions $h_{n,j}(z)$, whose coefficients are polynomials of degrees not greater than ${2m-2}$ in the functions $A_j(\mathbf{z}^{(m)})$ (with coefficients independent of $z$). The constant term of this polynomial in $h_{n,j}$ is equal to $D\bigl(A_0(\mathbf z^{(m)}), \dots,A_m(\mathbf z^{(m)})\bigr)$, the discriminant of the polynomial
Let us show that for $z\in\widehat{\mathbb{C}}\setminus F$ the zeros of the polynomial (29) are exactly equal to $f(\mathbf{z}^{(0)}), f(\mathbf{z}^{(1)}), \dots, f(\mathbf{z}^{(m-1)})$. Indeed, the value of the polynomial (29) at the point $f(\mathbf{z}^{(j)})$ is the product of the $j$th row of the matrix $A$ in (20) by the column $\bigl(A_0(\mathbf z^{(m)}),A_1(\mathbf z^{(m)}),\dots, A_m(\mathbf z^{(m)})\bigr)^\top$. On the other hand it is obvious from the definition (21) of the functions $A_j$ that this column is the last ($m$th) column of the matrix inverse to $A$. Therefore, for $j\neq m$ the product in question is zero. Hence for $z\in\widehat{\mathbb{C}}\setminus F$
$$
\begin{equation*}
D\bigl(A_0(\mathbf z^{(m)}), \dots,A_m(\mathbf z^{(m)})\bigr)=A_m^{2m-2}(\mathbf z^{(m)})\Pi_m(z).
\end{equation*}
\notag
$$
Since both sides of this equality are meromorphic in $\widehat{\mathbb{C}}\setminus F_m$, it is also true in $\widehat{\mathbb{C}}\setminus F_m$. Hence
$$
\begin{equation}
D_n(z)=R_n^{2m-2}(\mathbf z^{(m)})\bigl(A_m^{2m-2}(\mathbf z^{(m)})\Pi_m(z)+\Delta_n(z)\bigr),
\end{equation}
\tag{30}
$$
where
$$
\begin{equation*}
\Delta_n(z)=\sum_{\substack{0\leqslant j_0,j_1,\dots, j_m\leqslant 2m-2\\ 1\leqslant j_0+j_1+\dots+j_m\leqslant 2m-2}} M_{j_1,j_2,\dots,j_m}(z)h_{n,0}^{j_0}(z)\dotsb h_{n,m}^{j_m}(z),
\end{equation*}
\notag
$$
and the $M_{j_1,j_2,\dots,j_m}(z)$ are polynomials with integer coefficients of degrees not greater than $2m-2$ in the functions $A_j(\mathbf{z}^{(m)})$. Dividing $\Delta_n(z)$ by $A_m^{2m-2}(\mathbf{z}^{(m)})\Pi_m(z)$, we obtain (25). Proposition 1 is proved. Remark 1. Note that $\Pi_m(z)=(-1)^{m(m-1)/2}\widetilde\Pi_m(\mathbf{z}^{(m)})$, where Let us fix some conformal metric $\rho$ on $\mathfrak{R}$ and denote the distance in it by $\operatorname{dist}_\rho$ and the corresponding area form by $\sigma_\rho$. Since the $L^p$ spaces corresponding to the area forms of any two smooth positive Riemannian metrics on a compact Riemann surface coincide, we denote the $L^p$-space corresponding to the area form $\sigma_\rho$ by $L^p(\mathfrak{R})$. Also, for brevity, in what follows we use the notation $L^p(A):=L^p(A,d\sigma)$ for sets $A\subset\widehat{\mathbb{C}}$ (including the whole Riemann sphere $\widehat{\mathbb{C}}$). We will need some concepts and statements from potential theory on compact Riemann surfaces, which are presented in detail in [24], Appendix 2, and [11]. It is well known that the equation $\operatorname{dd^c}\widehat\nu=\nu$, where $\nu$ is a (real-valued) Borel signed measure with finite total variation on a compact Riemann surface $\mathcal{S}$, is solvable in currents if and only if the signed measure $\nu$ is neutral, that is, $\displaystyle\int_{\mathcal S}\nu=0$. The solution $\widehat\nu$ is called the potential of the signed measure $\nu$ and is defined up to addition of a harmonic function on $\mathcal{S}$, that is, a constant. The space of potentials of all neutral signed measures on $\mathcal{S}$ is precisely the space $\operatorname{\delta-sh}(\mathcal{S})$ of all $\delta$-subharmonic functions on $\mathcal{S}$ (that is, functions locally representable as a difference of two subharmonic functions). As is known, $\operatorname{\delta-sh}(\mathcal{S})\subset L^p(\mathcal{S})$ with respect to the area form of any smooth conformal metric on $\mathcal{S}$ for all $p\in[1,\infty)$. To eliminate the ambiguity in the definition of a potential, a continuous linear functional $\phi$ on $L^1(\mathcal{S})$ is usually fixed and the space
$$
\begin{equation}
\operatorname{Pot}_\phi(\mathcal S):=\{v\in\operatorname{\delta-sh}(\mathcal S)\colon \phi(v)=0\}
\end{equation}
\tag{32}
$$
is considered. Then for any neutral signed measure $\nu$ on $\mathcal{S}$ its $\phi$-normalized potential $\widehat\nu_\phi\in\operatorname{Pot}_\phi(\mathcal{S})$ is defined uniquely. We need two cases: (1) $\mathcal{S}=\widehat{\mathbb{C}}$; then we consider the functional
$$
\begin{equation}
\phi_1(v):=\int_{\widehat{\mathbb C}}v(z)\,d\sigma(z);
\end{equation}
\tag{33}
$$
(2) $\mathcal{S}=\mathfrak{R}$; then we consider the functional
$$
\begin{equation}
\phi_2(v):=\int_{\widehat{\mathbb C}\setminus F_m}v(\mathbf z^{(m)})\,d\sigma(z).
\end{equation}
\tag{34}
$$
In particular, the function $g(\mathbf{q},\mathbf{p};\mathbf{z})$ normalized by (16) is the $\phi_2$-normalized potential of the signed measure $2\pi(\delta_{\mathbf{q}}-\delta_{\mathbf{p}})$ on $\mathfrak{R}$, where $\delta_\mathbf{a}$ is the delta measure at the point $\mathbf{a}\in\mathfrak{R}$. To prove Theorems 3 and 4 we need the following lemma (Lemma 1), which gives an upper bound for the absolute values of all functions $h_{n,j}(z)$ from (24) in a neighbourhood of a fixed point $z_0\in\mathbb{C}\setminus F_m$. For any $r>0$ and $z\in\mathbb{C} $ we set $B_{z}^{r}=\{\xi\in\mathbb{C}\colon|\xi-z|<r\}$ (the Euclidean disc of radius $r$ with centre $z$). For any $a\in\widehat{\mathbb{C}}\setminus F_m$ we denote the maximum order of a pole of the functions $A_j(\mathbf{z})/A_j(\mathbf{z}^{(m)})$, $j=0,\dots,m$, at points in the set $\pi^{-1}(a)$ by $\Theta_0(a)$. We set $\Theta_0:=\max_{a\in\widehat{\mathbb{C}}\setminus F_m}\Theta_0(a)$. This maximum is finite since the $A_j(\mathbf{z})$ are meromorphic functions on $\mathfrak{R}$. Let ${\Theta=\Theta_0+M}$, where $M$ is the number of free zeros in the divisor of the remainder function $R_n$ from (17). Recall that, according to our notation, the $q_k(n)$ are projections of free zeros in the divisor of $R_n$. Lemma 1. Let $z_0\in\mathbb{C}\setminus F_m$. Then there exist numbers $\delta=\delta(z_0)\in(0,1)$ and $C=C(z_0)>0$ such that for $z\in B_{z_0}^{\delta}$ and all functions $h_{n,j}$ defined by (24), $j=0,1,\dots,m$ and $n\in\mathbb{N}$, the following estimate holds: Proof. Let the number $\delta$, $1>\delta>0$, satisfy the following conditions: $\overline{B_{z_0}^{2\delta}}\cap F_m=\varnothing$; there are no critical values of $\pi$ in $\overline{B_{z_0}^{2\delta}}\setminus z_0$; all functions $A_j(\mathbf{z})$, $j=0,\dots,m$, have neither zeros nor poles in the set $\pi^{-1}(\overline{B_{z_0}^\delta}\setminus z_0)$; the function $f(\mathbf{z})$ has no poles in the set $\pi^{-1}(B_{z_0}^{2\delta}\setminus z_0)$. The last condition also means $\pi^{-1}(B_{z_0}^{2\delta}\setminus z_0)$ contains no points $\mathbf{p}_k$ from the divisor of the remainder function $R_n$.
Substituting the expression (19) for $|R_n|$ into (24), for all $z\in\widehat{\mathbb{C}}\setminus F_m$ we obtain
$$
\begin{equation*}
\begin{aligned} \, |h_{n,j}(z)| &\leqslant\sum_{\mathbf z\in\pi^{-1}(z)\setminus\mathbf z^{(m)}} \biggl|\frac{A_j(\mathbf z)}{A_j(\mathbf z^{(m)})}\biggr|\, \biggl|\frac{R_n(\mathbf z)}{R_n(\mathbf z^{(m)})}\biggr| \\ &=\sum_{\mathbf z\in\pi^{-1}(z)\setminus\mathbf z^{(m)}}\biggl|\frac{A_j(\mathbf z)}{A_j(\mathbf z^{(m)})}\biggr| \frac{\psi_n(\mathbf z)}{\psi_n(\mathbf z^{(m)})} \exp(n(u(\mathbf z)-u(\mathbf z^{(m)}))). \end{aligned}
\end{equation*}
\notag
$$
The functions $(z-z_0)^{\Theta_0}A_j(\mathbf{z})/A_j(\mathbf{z}^{(m)})$ have no poles in the set $\pi^{-1}(\overline{B_{z_0}^\delta})$ and therefore are bounded in absolute value on this set by some constant $C_1$. Note also that for $z\in\widehat{\mathbb{C}}\setminus F_m$
$$
\begin{equation*}
u(\mathbf z^{(m)})-u(\mathbf z)\geqslant u_m(z)-u_{m-1}(z) \quad\text{as } \mathbf z\in\pi^{-1}(z)\setminus\mathbf z^{(m)}.
\end{equation*}
\notag
$$
Hence in $\overline{B_{z_0}^\delta}$, for all $j=0,\dots,m$
$$
\begin{equation}
|h_{n,j}(z)|\leqslant \frac{C_1\exp(-n(u_m(z)-u_{m-1}(z)))} {|z-z_0|^{\Theta_0}} \sum_{\mathbf z\in\pi^{-1}(z)\setminus\mathbf z^{(m)}}\frac{\psi_n(\mathbf z)}{\psi_n(\mathbf z^{(m)})}.
\end{equation}
\tag{36}
$$
To estimate the sum in the right-hand side of (36) we fix a point $\mathbf{p}^*\in\mathfrak{R}$ such that $|p^*-z_0|>2\delta$. Since for normalized (16) bipolar Green’s functions it is obvious that $g(\mathbf{q},\mathbf{p};\mathbf{z}) =g(\mathbf{q};\mathbf{p}^*;\mathbf{z})-g(\mathbf{p},\mathbf{p}^*;\mathbf{z})$, we see that
$$
\begin{equation}
\begin{aligned} \, \notag \frac{\psi_n(\mathbf z)}{\psi_n(\mathbf z^{(m)})} &=\exp\biggl\{\sum_{k=1}^M g(\mathbf q_k(n),\mathbf p^*;\mathbf z)- \sum_{k=1}^M g(\mathbf p_k,\mathbf p^*;\mathbf z) \\ &\qquad\qquad -\sum_{k=1}^M g(\mathbf q_k(n),\mathbf p^*;\mathbf z^{(m)})+ \sum_{k=1}^M g(\mathbf p_k,\mathbf p^*;\mathbf z^{(m)})\biggr\}. \end{aligned}
\end{equation}
\tag{37}
$$
Now let us estimate the bipolar Green’s functions $g$ occurring in each of the four sums in the right-hand side of (37), for $z\in B_{z_0}^\delta$. Note that, although for (36) we need to obtain such an estimate for $\mathbf{z}\in\pi^{-1}(B_{z_0}^\delta)\setminus \mathbf{z}^{(m)}$, we obtain it for all $\mathbf{z}\in\pi^{-1}(B_{z_0}^\delta)$.
Since there are no critical values of $\pi$ in $\overline{B_{z_0}^{2\delta}}\setminus z_0$, we see that $\pi^{-1}(B_{z_0}^{2\delta})=\bigsqcup_{l=0}^{m'}\mathbf{O}_l$, where $m'\leqslant m$ and the $\mathbf{O}_l$, $l=0,\dots,m'$, are connected open sets with pairwise disjoint closures such that each of them contains a unique point $\mathbf{z}_{0,l}$ from the set $\pi^{-1}(z_0)$. We assume that $\mathbf{z}_0^{(m)}=\mathbf{z}_{0,m'}\in\mathbf{O}_{m'}$. Let
$$
\begin{equation*}
s:=\min\Bigl\{\operatorname{dist}_\rho\bigl(\pi^{-1}(\overline{B_{z_0}^{\delta}}),\pi^{-1}(\widehat{\mathbb C}\setminus B_{z_0}^{2\delta})\bigr),\min_{0\leqslant l<l'\leqslant m'}\operatorname{dist}_\rho(\overline{\mathbf O_l},\overline{\mathbf O_{l'}})\Bigr\}.
\end{equation*}
\notag
$$
According to Corollary 6 in [24], there exists a positive constant $C_s$ such that
$$
\begin{equation}
|g(\mathbf q,\mathbf p;\mathbf z)|\leqslant C_s \quad\text{if } \operatorname{dist}_\rho(\mathbf q,\mathbf z)\geqslant s \text{ and } \operatorname{dist}_\rho(\mathbf p,\mathbf z)\geqslant s.
\end{equation}
\tag{38}
$$
Let $\mathbf{z}\in\pi^{-1}(B_{z_0}^\delta)$. First we deduce upper and lower estimates for the functions $g(\mathbf{p}_k,\mathbf{p}^*;\mathbf{z})$. So we will estimate the functions $g$ in the second and fourth sums in the right-hand side of (37). Since we have chosen $\mathbf{p}^*$ so that $\mathbf{p}^*\in \pi^{-1}(\widehat{\mathbb{C}}\setminus B_{z_0}^{2\delta})$, we have $\operatorname{dist}_\rho(\mathbf{z},\mathbf{p}^*)\geqslant s$ for all $\mathbf{z}\in\pi^{-1}(B_{z_0}^\delta)$. Let $\mathbf{z}\in\pi^{-1}(B_{z_0}^\delta)\cap\mathbf{O}_l$ for some $l=0,\dots,m'$. If $\mathbf{p}_k\notin\mathbf{O}_l$, then $\operatorname{dist}_\rho(\mathbf{p}_k,\mathbf{z})\geqslant s$, and we have $|g(\mathbf{p}_k,\mathbf{p}^*;\mathbf{z})|\leqslant C_s$ by (38). Now let $\mathbf{p}_k\in\mathbf{O}_l$. Then $\mathbf{p}_k=\mathbf{z}_{0,l}$ by the choice of $\delta$. Let $d$ be the order of $\mathbf{z}_{0,l}$ as a critical point of $\pi$ (if $\mathbf{z}_{0,l}$ is not a critical point of $\pi$, then $d=1$). Let $\zeta$ be a holomorphic coordinate in $\mathbf{O}_l$ such that $z=z_0+\zeta^d(\mathbf{z})$ for $\mathbf{z}\in\mathbf{O}_l$. Then $\zeta(\mathbf{p}_k)=0$ and from the definition (15) of bipolar Green’s functions we conclude that the difference $g(\mathbf{p}_k,\mathbf{p}^*;\mathbf{z})-\log|\zeta(\mathbf{z})|$ is a harmonic function in $\mathbf{O}_l$. Therefore, there exists a positive constant $\widetilde C_l $ such that $\bigl|g(\mathbf p_k,\mathbf p^*;\mathbf z)-\log|\zeta(\mathbf z)|\bigr|\leqslant \widetilde C_l$ in $\mathbf{O}_l\cap\pi^{-1}(\overline{B_{z_0}^\delta})$. Since $|\zeta(\mathbf{z})|<\delta^{1/d}<1$ for $z\in B_{z_0}^\delta$ and $z=z_0+\zeta^d(\mathbf{z})$ for $\mathbf{z}\in\mathbf{O}_l$, we obtain
$$
\begin{equation*}
\begin{aligned} \, -\widetilde C_l+\log|z-z_0| &=-\widetilde C_l+d\log|\zeta(\mathbf z)|\leqslant -\widetilde C_l+\log|\zeta(\mathbf z)|\leqslant g(\mathbf p_k,\mathbf p^*;\mathbf z) \\ &\leqslant \widetilde C_l+\log|\zeta(\mathbf z)|\leqslant \widetilde C_l. \end{aligned}
\end{equation*}
\notag
$$
Combining such estimates in all the $\mathbf{O}_l$, for $\mathbf{z}\in\pi^{-1}(B_{z_0}^\delta)$ we obtain
$$
\begin{equation}
-c_1+\log|z-z_0|\leqslant g(\mathbf p_k,\mathbf p^*;\mathbf z)\leqslant c_1,
\end{equation}
\tag{39}
$$
where $c_1 = \max\{C_s,\widetilde C_0,\dots,\widetilde C_{m'}\}$.
Now we find lower estimates of $g(\mathbf{q},\mathbf{p}^*;\mathbf{z}^{(m)})$ for $z\in B_{z_0}^\delta$ and thus lower estimates of the functions $g$ occurring in the third sum in the right-hand side of (37). Since $z\in B_{z_0}^\delta$, we have $\mathbf{z}^{(m)}\in\mathbf{O}_{m'}\cap B_{z_0}^\delta$. If $\mathbf{q}\notin\mathbf{O}_{m'}$, then $\operatorname{dist}_\rho(\mathbf{q},\mathbf{z})\geqslant s$ and by (38) we have $|g(\mathbf{q},\mathbf{p}^*;\mathbf{z})|\leqslant C_s$. Now let $\mathbf{q}\in\mathbf{O}_{m'}$. Note that, since $z_0\notin F_m$, we see that $\mathbf{z}_{0,m'}=\mathbf{z}_0^{(m)}$ is not a critical point of the projection $\pi$, and so $\pi$ maps $\mathbf{O}_{m'}$ biholomorphically onto $B_{z_0}^{2\delta}$. Consequently, $z$ is a coordinate in $\mathbf{O}_{m'}$. By the definition of the bipolar Green’s functions (15) the difference $g(\mathbf{q},\mathbf{p}^*;\mathbf{z}^{(m)})-\log|z-q|$, considered as a function of $z$ in the disc $B_{z_0}^{2\delta+\varepsilon}$ for a sufficiently small fixed $\varepsilon>0$, is harmonic in this disc. Denoting the Lebesgue measure on $\mathbb{C}$ by $\sigma_L$ and applying the mean value theorem in the disc $\overline{B_{z}^\delta}$, we obtain
$$
\begin{equation}
\begin{aligned} \, \notag &\bigl|g(\mathbf q,\mathbf p^*;\mathbf z^{(m)})-\log|z-q|\bigr| =\biggl|\frac1{\pi\delta^2}\int_{\overline{B_{z}^\delta}}\bigl(g(\mathbf q,\mathbf p^*; {\xi}^{(m)})-\log|\xi-q|\bigr)\,d\sigma_L(\xi)\biggr| \\ &\qquad\leqslant \frac1{\pi\delta^2}\int_{\overline{B_{z}^\delta}}|g(\mathbf q,\mathbf p^*; {\xi}^{(m)})|\,d\sigma_L(\xi)+ \frac1{\pi\delta^2}\int_{\overline{B_{z}^\delta}}\bigl|\log|\xi-q|\bigr|\,d\sigma_L(\xi). \end{aligned}
\end{equation}
\tag{40}
$$
Clearly, $\overline{B_z^\delta}\subset \overline{B_{z_0}^{2\delta}}$ for all $z\in B_{z_0}^\delta$. Therefore, for the first of the integrals on the right-hand side of (40) we have the inequality
$$
\begin{equation}
\int_{\overline{B_{z}^\delta}}|g(\mathbf q,\mathbf p^*; {\xi}^{(m)})|\,d\sigma_L(\xi) \leqslant \int_{\overline{B_{z_0}^{2\delta}}}|g(\mathbf q,\mathbf p^*; {\xi}^{(m)})|\,d\sigma_L(\xi).
\end{equation}
\tag{41}
$$
Since $z$ is a coordinate in some small neighbourhood of the closure $\overline{\mathbf{O}_{m'}}$, it is obvious that the measure $\sigma_L$ has a smooth density with respect to the measure $\sigma_\rho$ on $\overline{\mathbf{O}_{m'}}$, which is therefore bounded by some constant $C_2$, and so
$$
\begin{equation}
\int_{\overline{B_{z_0}^{2\delta}}}|g(\mathbf q,\mathbf p^*; {\xi}^{(m)})|\,d\sigma_L(\xi)\leqslant C_2 \int_{\overline{\mathbf O_{m'}}}|g(\mathbf q,\mathbf p^*; {\xi})|\,d\sigma_\rho( {\xi})\leqslant C_2\|g(\mathbf q,\mathbf p^*;\,\cdot\,)\|_{L_1(\mathfrak R)}.
\end{equation}
\tag{42}
$$
From Corollary 5 in [24] it follows that the norms $\|g(\mathbf{q},\mathbf{p};\,\cdot\,)\|_{L_1(\mathfrak{R})}$ are uniformly bounded for all $\mathbf{q},\mathbf{p}\in\mathfrak{R}$ by some constant $C_\rho$. Now we estimate the second term in the right-hand side of (40). Since $z\in B_{z_0}^{\delta}$ and $q\in\overline{B_{z_0}^{2\delta}}$, we see that $|z-q|\leqslant 3\delta$, and therefore
$$
\begin{equation}
\int_{\overline{B_{z}^\delta}}\bigl|\log|\xi-q|\bigr|\,d\sigma_L(\xi)\leqslant \int_{\overline{B_q^{4\delta}}}\bigl|\log|\xi-q|\bigr|\,d\sigma_L(\xi)= \int_{\overline{B_0^{4\delta}}}\bigl|\log|\xi|\bigr|\,d\sigma_L(\xi)=C_3.
\end{equation}
\tag{43}
$$
Thus, from (40), taking (41)–(43) into account, for $z\in B_{z_0}^\delta$ and $\mathbf{q}\in\mathbf{O}_{m'}$ we obtain
$$
\begin{equation}
g(\mathbf q,\mathbf p^*;\mathbf z^{(m)}) \geqslant-c_2+\log|z-q|,
\end{equation}
\tag{44}
$$
where $c_2:=(C_2C_\rho+C_3)/\pi\delta^2$. Since $0<\delta<1$ and for $\mathbf{q}\notin\mathbf{O}_{m'}$ we have $|g(\mathbf{q},\mathbf{p}^*;\mathbf{z})|\leqslant C_s$, from (44) for $z\in B_{z_0}^\delta$ and an arbitrary $\mathbf{q}\in\mathfrak{R}$ we obtain
$$
\begin{equation}
g(\mathbf q,\mathbf p^*;\mathbf z^{(m)}) \geqslant-c_2-C_s+\min\{\log|z-q|, \log\delta\}.
\end{equation}
\tag{45}
$$
Finally, to estimate the functions $g$ in the first sum on the right-hand side of (37) we show that for all $\mathbf{q}\in\mathfrak{R}$ and $z\in B_{z_0}^\delta$ the functions $g(\mathbf{q},\mathbf{p}^*;\mathbf{z})$ are uniformly bounded by a constant. Let $\mathbf{z}\in\pi^{-1}(B_{z_0}^\delta)\cap\mathbf{O}_l$ for some $l=0,\dots,m'$. Again, if $\mathbf{q}\notin\mathbf{O}_l$, then $|g(\mathbf{q},\mathbf{p}^*;\mathbf{z})|\leqslant C_s$. Let $\mathbf{q}\in\mathbf{O}_l$. First consider the case when $l=m'$. Then from (40), taking (41)–(43) into account, for $z\in B_{z_0}^\delta$ and $\mathbf{q}\in\mathbf{O}_{m'}$ we obtain
$$
\begin{equation}
g(\mathbf q,\mathbf p^*;\mathbf z^{(m)})\leqslant c_2+\log|z-q|\leqslant c_2+\log 3.
\end{equation}
\tag{46}
$$
Now let $l$ be an arbitrary integer number from $0$ to $m'$. Again, let $d$ be the order of $\mathbf{z}_{0,l}$ as a critical point of $\pi$, and let $\zeta$ be a holomorphic coordinate in $\mathbf{O}_l$ such that $z=z_0+\zeta^d(\mathbf{z})$ for $\mathbf{z}\in\mathbf{O}_l$. Then estimates similar to (40)–(43), but in the $\zeta$-coordinate hold for $\mathbf{z}\in\pi^{-1}(B_{z_0}^\delta)\cap\mathbf{O}_l$, provided that the circles of the integration are appropriately chosen. Therefore, for $\mathbf{z}\in\pi^{-1}(B_{z_0}^\delta)\cap\mathbf{O}_l$ and $\mathbf{q}\in\mathbf{O}_{l}$ the following estimate is valid:
$$
\begin{equation}
g(\mathbf q,\mathbf p^*;\mathbf z)\leqslant \widetilde c_l+\log|\zeta(\mathbf z)-\zeta(\mathbf q)|\leqslant \widetilde c_l+\log 3,
\end{equation}
\tag{47}
$$
where $\widetilde c_l$ is some constant. Setting $c_3:=\max\{C_s,\widetilde c_0+\log 3,\dots,\widetilde c_{m'}+\log 3\}$, from (47), for all $\mathbf{z}\in B_{z_0}^\delta$ and an arbitrary $\mathbf{q}\in\mathfrak{R}$ we obtain
Combining estimates (39), (45) and (48), from (37) we derive
$$
\begin{equation}
\frac{\psi_n(\mathbf z)}{\psi_n(\mathbf z^{(m)})}\leqslant \exp(Mc_3) \frac{\exp(Mc_1)}{|z-z_0|^M} \frac{\exp(M(c_2+C_s))}{\prod_{k=1}^{M} \min\{|z-q_k(n)|,\delta\}} \exp(Mc_1).
\end{equation}
\tag{49}
$$
Substituting the estimate (49) into (36), we obtain (35) for $C=mC_1\exp(M(2c_1+c_2+c_3+C_s))$. Lemma 1 is proved. § 4. The proof of Theorem 2The methods of the proof of Theorem 2 are similar to those used in the proof of Theorem 1 given in [24], Theorems 1 and 2, and rely on representation (25). We present detailed schemes of the proof and refer to [24] for details and necessary estimates. 4.1. The proof of part (1)Recall that $M$ is the number of free zeros of the remainder function $R_n$. Let $B$ be the difference between the numbers of zeros and poles (taking account of multiplicities) of the function $A_m^{2m-2}(\mathbf{z}^{(m)})\Pi_m(z)$ in ${\widehat{\mathbb{C}}\setminus F_m}$, which are both finite (see Remark 1). Part (1) of Theorem 2 follows from Statement 1 proved below. Statement 1. For any neighbourhood $V$ of the compact set $F_m$ there exists ${N=N(V)}$ such that for all $n>N$ at most $L':=(2m-2)M+B$ zeros of the discriminant $D_n(z)$ lie outside $V$. Proof. Reducing the neighbourhood $V$ if necessary, we can assume that the functions $A_j(\mathbf{z}^{(m)})$, $j=0,\dots,m$, and $\Pi_m(z)$ have neither zeros nor poles in $V\setminus F_m$. In the case when $\infty\notin F_m$ we also assume that $\infty\notin V$.
As in the proof of Statement 1 in [24], we set
$$
\begin{equation*}
\delta:=\frac{\operatorname{dist}(\partial V,F_m)}{2(2M+3)}
\end{equation*}
\notag
$$
and for each $n$ we choose a system of disjoint smooth contours $\Gamma_n$ such that $\Gamma_n$ bounds an open set $\mathrm{D}_n$, where $F_m\subset \mathrm{D}_n\subset V$ and the following conditions are satisfied:
$$
\begin{equation}
\begin{gathered} \, \notag \operatorname{dist}(\Gamma_n,F_m)\geqslant \delta, \qquad \operatorname{dist}(\Gamma_n,\partial V)\geqslant \delta, \\ \operatorname{dist}(\Gamma_n,p_k)\geqslant \delta \quad\text{and}\quad \operatorname{dist}(\Gamma_n,q_k(n))\geqslant\delta \quad\text{for } k=1,\dots, M, \end{gathered}
\end{equation}
\tag{50}
$$
where $p_k=\pi(\mathbf{p}_k)$ and $q_k(n)=\pi(\mathbf{q}_k(n))$ are the projections of poles and zeros of the function $R_n$. We also define the compact set
$$
\begin{equation*}
K:= \biggl\{z\in V\colon \operatorname{dist}(z,F_m)\geqslant \frac{\delta}{2},\ \operatorname{dist}(z,\partial V)\geqslant \frac{\delta}{2}\biggr\}.
\end{equation*}
\notag
$$
Let us obtain an upper estimate for the number of zeros of the discriminant $D_n(z)$ in $\Omega_n:=\widehat{\mathbb{C}}\setminus \overline{\mathrm{D}}_n$ (and so also in $\widehat{\mathbb{C}}\setminus V$) using representation (25). Since the functions $A_j(\mathbf{z}^{(m)})$ and $\Pi_m(z)$ have neither zeros nor poles on $K$, we see that their absolute values, as well as the absolute values of the functions $M_{j_1,j_2,\dots, j_m}(z)$ in (26), are bounded and separated from zero on $K$. In the proof of Statement 1 in [24] it was shown that $\lim_{n\to\infty}\max_{z\in\Gamma_n}|h_{n,j}(z)|=0$. Consequently, for the functions $H_n(z)$ defined in (26) we have $\lim_{n\to\infty}\max_{z\in\Gamma_n}|H_n(z)|=0$. This means that there exists $N$ such that for all $n>N$ we have $|H_n(z)|<1/2$ on $\Gamma_n$. Throughout the rest of the proof of the statement we assume that $n>N$. Then the function $1+H_n(z)$ has no zeros on $\Gamma_n$. This means that the polynomial $D_n(z)$ has no zeros on $\Gamma_n$ by (25), and to calculate the number of its zeros in $\Omega_n$ we can use the argument principle. We orient each of the contours making up $\Gamma_n$ in the positive direction with respect to $\Omega_n$. Since $|H_n(z)|<1/2$ on $\Gamma_n$, we have $\operatorname{\Delta}_{z\in\Gamma_n}\arg(1+H_n(z))=0$. Hence, by (25) we have
$$
\begin{equation}
\operatorname*{\Delta}_{z\in\Gamma_n}\arg D_n(z)= (2m-2)\operatorname*{\Delta}_{z\in\Gamma_n}\arg R_n(\mathbf z^{(m)})+\operatorname*{\Delta}_{z\in\Gamma_n}\arg \bigl(A_m^{2m-2}(\mathbf z^{(m)})\Pi_m(z)\bigr).
\end{equation}
\tag{51}
$$
Here $R_n(\mathbf{z}^{(m)})$ and $A_m(\mathbf{z}^{(m)})$ are understood as meromorphic functions of $z\in\widehat{\mathbb{C}}\setminus F_m$, that is, for example, $R_n(\mathbf{z}^{(m)})=R_n \circ (\pi|_{\mathfrak{R}^{(m)}})^{-1}(z)$. Since the function $A_m^{2m-2}(\mathbf{z}^{(m)})\Pi_m(z)$ has neither zeros nor poles in $V\setminus F_m$, we see that the second term in (51) equals $2\pi B$ by the argument principle.
First consider the case when $\infty\notin F_m$. Taking into account the form (17) of the divisor $(R_n)$ and the fact that $\infty\in \Omega_n$, we can conclude that the difference between the number of zeros and the number of poles (taking account of multiplicities) of the function $R_n(\mathbf{z}^{(m)})$ in $\Omega_n$ does not exceed $-n+M$, so that $(2\pi)^{-1}\operatorname{\Delta}_{z\in\Gamma_n}\arg R_n(\mathbf{z}^{(m)})\leqslant -n+M$. Substituting this estimate into (51) we have
$$
\begin{equation*}
\frac{1}{2\pi}\operatorname*{\Delta}_{z\in\Gamma_n}\arg D_n(z)\leqslant (2m-2)(-n+M)+B.
\end{equation*}
\notag
$$
Since $\infty\in\Omega_n$ and $\deg D_n\leqslant (2m-2)n$, by the argument principle the number of zeros of $D_n$ in $\Omega_n$ is
$$
\begin{equation*}
\operatorname{deg} D_n(z)+\frac{1}{2\pi}\operatorname*{\Delta}_{z\in\Gamma_n}\arg D_n(z)\leqslant (2m-2)M+B=L'.
\end{equation*}
\notag
$$
Now let $\infty\in F_m$. Then we conclude from the form (17) of the divisor $(R_n)$ that the number of zeros of the function $R_n(\mathbf{z}^{(m)})$ in $\widehat{\mathbb{C}}\setminus F_m$ does not exceed $M$. Therefore,
$$
\begin{equation*}
\frac{1}{2\pi}\operatorname*{\Delta}_{z\in\Gamma_n}\arg R_n(\mathbf z^{(m)})\leqslant M.
\end{equation*}
\notag
$$
Since $\infty\notin \Omega_n$ for all $n$, we see that the number of zeros of $D_n(z)$ in $\Omega_n$ is
$$
\begin{equation*}
\frac{1}{2\pi}\operatorname*{\Delta}_{z\in\Gamma_n}\arg D_n(z)
\end{equation*}
\notag
$$
and from (51) we obtain that this number does not exceed $(2m-2)M+B=L'$.
Statement 1 is proved. To prove the other parts of Theorem 2, we need the following simple proposition about zeros and poles in $\widehat{\mathbb{C}}\setminus F_m$ of the function $1+H_n$ from Proposition 1. Let $\widetilde B$ be the number of zeros (taking account of multiplicities) of the function $A_m^{2m-2}(\mathbf{z}^{(m)})\Pi_m(z)$ in $\widehat{\mathbb{C}}\setminus F_m$. Proposition 2. (1) The number of poles (taking account of multiplicities) of the function $1+H_n(z)$ in $\widehat{\mathbb{C}}\setminus F_m$ does not exceed $\widetilde L:=(2m-2)M+\widetilde B$. (2) For any neighbourhood $V$ of the compact set $F_m$ there exists $N=N(V)$ such that for $n>N$ the functions $1+H_n(z)$ have at most $\widetilde L$ zeros in $\widehat{\mathbb{C}}\setminus V$ (taking account of multiplicities). Proof. From (25), in $\widehat{\mathbb{C}}\setminus F_m$ we obtain Taking into account the form of the divisor $(R_n)$, we immediately obtain part (1).
To prove (2), for each $n$ consider the system of contours $\Gamma_n$ constructed in the proof of Statement 1. For each $n$ this system of contours $\Gamma_n$ bounds an open set $\mathrm{D}_n$ such that $F_m\subset \mathrm{D}_n\subset V$. In the proof of Statement 1 it was shown that there exists $N=N(V)$ such that $|H_n(z)|\leqslant1/2$ for $z\in\Gamma_n$ for all $n>N$. Thus, by the argument principle the difference between the number of zeros and the number of poles of $1+H_n$ in the domain $\Omega_n:=\widehat{\mathbb{C}}\setminus\overline{\mathrm{D}}_n$ is equal to $(2\pi)^{-1}\operatorname{\Delta}_{z\in\Gamma_n}\arg(1+H_n(z))=0$. Consequently, the number of zeros of $1+H_n(z)$ in $\Omega_n$, and therefore in $\widehat{\mathbb{C}}\setminus V$, does not exceed the number of its poles in $\widehat{\mathbb{C}}\setminus F_m$, which in turn does not exceed $\widetilde L$. Proposition 2 is proved. 4.2. Proofs of parts (2) and (3) of Theorem 2Fix $p\in[1;\infty)$. From representation (25) of the function $D_n$, taking (19) into account we conclude that for ${z\in \widehat{\mathbb{C}}\setminus F_m}$
$$
\begin{equation}
\begin{aligned} \, \notag &\frac1{(2m-2)n}\log|D_n(z)|=u(\mathbf z^{(m)})+\frac1{n}\log|\psi_n(\mathbf z^{(m)})|+ \frac1{n}\log|A_m(\mathbf z^{(m)})| \\ &\qquad\qquad +\frac1{(2m-2)n}\log|\Pi_m(z)|+\frac1{(2m-2)n}\log|1+H_n(z)|. \end{aligned}
\end{equation}
\tag{52}
$$
Since $F_m$ is a one-dimensional piecewise analytic subset of $\widehat{\mathbb{C}}$ (see [24], Lemma 3), we see that the area $\sigma(F_m)$ is zero. Therefore, it is obvious from the definitions (18) and (5) of the functions $\psi_n(\mathbf{z})$ and $u(\mathbf{z})$ that the functions $\log|\psi_n(\mathbf{z}^{(m)})|$ and $u(\mathbf{z}^{(m)})$ defined formally for $z\in \widehat{\mathbb{C}}\setminus F_m$ belong to $L^p(\widehat{\mathbb{C}})$. Since $\Pi_m(z)=\widetilde\Pi_m(\mathbf{z}^{(m)})$ and the functions $A_m(\mathbf{z})$ and $\widetilde\Pi_m(\mathbf{z})$ are meromorphic on $\mathfrak{R}$, the functions $\log |A_m(\mathbf{z}^{(m)})|$ and $\log|\Pi_m(z)|$ also belong to $L^p(\widehat{\mathbb{C}})$. Since $D_n(z)$ is a polynomial, we have $\log|D_n(z)|\in L^p(\widehat{\mathbb{C}})$. Therefore, we conclude from (52) that $\log|1+H_n(z)|\in L^p(\widehat{\mathbb{C}})$. In [24], Corollary 5, it was shown that the bipolar Green’s functions $g(\mathbf{q},\mathbf{p};\mathbf{z})$ which are spherically normalized on the $m$th sheet are bounded in $L^p(\mathfrak{R})$ uniformly for all $\mathbf{q},\mathbf{p}\in\mathfrak{R}$. Hence it obviously follows in view of representation (18) that $n^{-1}\log|\psi_n(\mathbf{z}^{(m)})|\to 0$ in $L^p(\widehat{\mathbb{C}})$ as $n\to \infty$. We now prove that $n^{-1}\log|1+H_n(z)|\to 0$ in $L^p_{\mathrm{loc}}(\widehat{\mathbb{C}}\setminus F_m)$ as $n\to\infty$. Given the above and the fact that $u(\mathbf{z}^{(m)})=u_m(z)$ in $L^p(\widehat{\mathbb{C}})$, it follows that $((2m-2)n)^{-1}\log|D_n(z)|\to u_m(z)$ in $L^p_{\mathrm{loc}}(\widehat{\mathbb{C}}\setminus F_m)$. Thus, we fix a neighbourhood $V$ of the compact set $F_m$ and show that $n^{-1}\log|1+H_n(z)|\to 0$ in $L^p(\widehat{\mathbb{C}}\setminus V)$. Reducing the neighbourhood $V$ if necessary, we can assume that the functions $A_j(\mathbf{z}^{(m)})$, $j=0,\dots,m$, and $\Pi_m(z)$ have neither zeros nor poles in $V\setminus F_m$ and, moreover, if $\infty\notin F_m$, then $\infty\notin V$. Set
$$
\begin{equation}
\delta:=\frac{\operatorname{dist}(\partial V, F_m)}{2(2M+2\widetilde L+3)}.
\end{equation}
\tag{53}
$$
Let $V_{\delta}:=\{z\in\widehat{\mathbb{C}}\colon\operatorname{dist}(z,\partial V)<\delta\}$. According to Proposition 2, there exists $N=N(V_{\delta})$ such that for $n>N$ all functions $1+H_n(z)$ have at most $\widetilde L$ zeros and poles in $\widehat{\mathbb{C}}\setminus V_{\delta}$. Further, we assume that $n>N$. Let $\widetilde q_1(n), \widetilde q_2(n), \dots, \widetilde q_{l(n)}(n)$, $l(n)\leqslant \widetilde L$, be the zeros of the function $1+H_n(z)$ in $\widehat{\mathbb{C}}\setminus V_{\delta}$ (written out taking account of multiplicities), and let $\widetilde p_1(n), \widetilde p_2(n), \dots, \widetilde p_{l'(n)}(n)$, $l'(n)\leqslant \widetilde L$, be its poles in $\widehat{\mathbb{C}}\setminus V_{\delta}$ (also written out taking account of multiplicities). Let
$$
\begin{equation*}
\widetilde{\mathbf q}_s(n):=(\pi|_{\mathfrak R^{(m)}})^{-1}(\widetilde q_s(n)) \quad\text{and}\quad \widetilde{\mathbf p}_s(n):=(\pi|_{\mathfrak R^{(m)}})^{-1}(\widetilde p_s(n)).
\end{equation*}
\notag
$$
Fix a point $\mathbf{z}^*\in\partial\mathfrak{R}^{(m)}$ and set
$$
\begin{equation}
\widetilde\psi_n(\mathbf z):=\exp\biggl\{\sum_{s=1}^{\widetilde L}g(\widetilde{\mathbf q}_k(n),\mathbf z^*;\mathbf z)+\sum_{s=1}^{\widetilde L}g(\mathbf z^*,\widetilde{\mathbf p}_k(n);\mathbf z)\biggr\}, \qquad\mathbf z\in\mathfrak R,
\end{equation}
\tag{54}
$$
where $q(\mathbf{q},\mathbf{p};\mathbf{z})$ are bipolar Green’s functions (normalized by condition (16)). We assume in (54) that $l(n)=l'(n)=\widetilde L$, supplementing the tuples $\{\widetilde{\mathbf{q}}_s(n)\}_{s=1}^{l(n)}$ and $\{\widetilde{\mathbf{p}}_s(n)\}_{s=1}^{l'(n)}$ with the point $\mathbf{z}^*$ taken the required number of times if necessary, and adding the point $z^*:=\pi(\mathbf{z}^*)$ to the tuples $\{\widetilde q_s(n)\}_{s=1}^{l(n)}$ and $\{\widetilde p_s(n)\}_{s=1}^{l'(n)}$ the corresponding number of times. As already noted, it was shown in [24], Corollary 5, that the functions $g(\mathbf{q},\mathbf{p};\mathbf{z})$ are uniformly bounded in $L^p(\mathfrak{R})$ for all $\mathbf{q},\mathbf{p}\in\mathfrak{R}$, and therefore $n^{-1} \log\widetilde\psi_n(\mathbf{z}^{(m)})\to 0$ in $L^p(\widehat{\mathbb{C}})$ as $n\to\infty$. Thus, it remains to show that
$$
\begin{equation}
\frac 1 n \log\frac{|1+H_n(z)|}{\widetilde\psi_n(\mathbf z^{(m)})}\to0 \quad\text{in } L^p(\widehat{\mathbb C}\setminus V) \quad\text{as } n\to\infty.
\end{equation}
\tag{55}
$$
To do this we prove that the functions $\log(|1+H_n(z)|/\widetilde\psi_n(\mathbf{z}^{(m)}))$ are uniformly bounded on the set $\widehat{\mathbb{C}}\setminus V$. By the choice of $\delta$ (53), for each $n$ one can choose a system of disjoint smooth contours $\Gamma_n$ such that $\Gamma_n$ bounds an open set $\mathrm{D}_n$, where $F_m\subset\mathrm{D}_n\subset V$, and the following conditions are satisfied: $\operatorname{dist}(\Gamma_n,F_m)\geqslant \delta$, $\operatorname{dist}(\Gamma_n,\partial V)\geqslant \delta$, $\operatorname{dist}(\Gamma_n,\widetilde p_s(n))\geqslant \delta$ and $\operatorname{dist}(\Gamma_n,\widetilde q_s(n))\geqslant \delta$ for $s=1,\dots,\widetilde L$, $\operatorname{dist}(\Gamma_n,p_k)\geqslant\delta$ and $\operatorname{dist}(\Gamma_n,q_k(n))\geqslant\delta$ for $k=1,\dots, M$, where $p_k=\pi(\mathbf{p}_k)$ and $q_k(n)=\pi(\mathbf{q}_k(n))$ are the projections of poles and zeros of the function $R_n$, respectively (see (19)). In particular, the system of contours $\Gamma_n$ satisfies conditions (50), so, as shown in the proof of Statement 1, there exists $N'$ such that for $n>N'$ the condition $|H_n(z)|<1/2$ is satisfied on $\Gamma_n$. Throughout the proof we assume that $n>N'$ (and $n>N$). Then $1/2<|1+H_n(z)|<3/2$ for $z\in\Gamma_n$. On the other hand, since $\operatorname{dist}(\Gamma_n,\widetilde p_s(n))\geqslant\delta$ and $\operatorname{dist}(\Gamma_n,\widetilde q_s(n))\geqslant\delta$, and also $\operatorname{dist}(\Gamma_n,z^*)\geqslant\delta$, by Corollary 6 in [24] there exists a constant $C=C(\delta)$ such that for $z\in\Gamma_n$ the inequalities $|g(\widetilde{\mathbf{q}}_s(n),\mathbf{z}^*;\mathbf{z}^{(m)})|\leqslant C$ and $|g(\mathbf{z}^*,\widetilde{\mathbf{p}}_s(n);\mathbf{z}^{(m)})|\leqslant C$ hold for $s=1,\dots,\widetilde L$. Therefore, from the definition of the functions $\widetilde\psi_n$ we obtain $\exp(-2\widetilde LC)\leqslant\widetilde\psi_n(\mathbf{z}^{(m)})\leqslant \exp(2\widetilde LC)$ for $z\in\Gamma_n$. Thus, for $z\in\Gamma_n$
$$
\begin{equation}
\biggl|\log\frac{|1+H_n(z)|}{\widetilde\psi_n(\mathbf z^{(m)})}\biggr|\leqslant 2\widetilde LC+1.
\end{equation}
\tag{56}
$$
Since $H_n(z)$ is meromorphic in $\widehat{\mathbb{C}}\setminus F_m$, we see that $\log|1+H_n|$ is a harmonic function in $\widehat{\mathbb{C}}\setminus F_m$ away from the zeros and poles of $1+H_n$, at which $\log|1+H_n|$ has logarithmic singularities. In $\widehat{\mathbb{C}}\setminus V_\delta$ these singularities are only at the points $\widetilde q_s(n)$, $s=1,\dots, l(n)$, and $\widetilde p_s(n)$, $s=1,\dots, l'(n)$. By the definition (54) of $\widetilde\psi_n$ the function $\log\widetilde\psi_n(\mathbf{z}^{(m)}$) is harmonic in $\widehat{\mathbb{C}}\setminus F_m$ away from the points $\widetilde q_s(n)$, $s=1,\dots, l(n)$, and $\widetilde p_s(n)$, $s=1,\dots, l'(n)$, at which it has precisely the same logarithmic singularities as $\log|1+H_n(z)|$. Therefore, the function $\log(|1+H_n(z)|/\widetilde\psi_n(\mathbf{z}^{(m)}))$ is harmonic in $\widehat{\mathbb{C}}\setminus V_{\delta}$. Since $\widehat{\mathbb{C}}\setminus V\subset\widehat{\mathbb{C}}\setminus \overline{\mathrm{D}}_n\subset\widehat{\mathbb{C}}\setminus V_{\delta}$ and $\Gamma_n=\partial(\widehat{\mathbb{C}}\setminus\overline{\mathrm{D}}_n)$, it follows from the maximum principle for harmonic functions that the estimate (56) is true for all $z\in\widehat{\mathbb{C}}\setminus V$ and (55) is proved. Thus, we have shown that $((2m-2)n)^{-1}\log|D_n(z)|\to u_m(z)$ in $L_{\mathrm{loc}}^p(\widehat{\mathbb{C}}\setminus F_m)$. To complete the proof of parts (2) and (3) of Theorem 2 we show that from any subsequence $\{D_n^*(z)\}$, $n\in\Lambda$, one can extract a subsequence $\{D_n^*(z)\}$, $n\in\Lambda'$, so that (13) and (14) hold for it. By the Poincaré–Lelong formula (see [12], for instance) we obtain
$$
\begin{equation*}
\mu_n:=\frac{1}{(2m-2)n}\operatorname{dd^c}\log|D_n^*|=\frac{2\pi}{(2m-2)n}\biggl(\sum_{D_n(z)=0}\delta_z-(\operatorname{deg} D_n^*)\cdot\delta_\infty\biggr),
\end{equation*}
\notag
$$
where $\delta_z$ is the delta measure at $z\in\widehat{\mathbb{C}}$. Since $\deg D_n^*\leqslant (2m-2) n$, we have $\|\mu_n\|_{C(\widehat{\mathbb{C}})^*}\leqslant 4\pi$. By the Banach–Alaoglu theorem on the compactness of a ball in the dual space in the $*$-weak topology, from the sequence $\{\mu_n\}$, $n\in\Lambda$, one can extract a subsequence $\{\mu_n\}$, $n\in\Lambda'$, that converges $*$-weakly to some signed measure $\mu\in C(\widehat{\mathbb{C}})^*$. Let us show that (13) and (14) hold for the subsequence $\{\mu_n\}$, $n\in\Lambda'$. Since the measure $d\sigma$ defining the functional $\phi_1$ from (33) has continuous local potentials (because it has a smooth density with respect to the Lebesgue measure in any coordinate neighbourhood), according to the corollary of the lemma in § 2.3 in [11], from the $*$-weak convergence of $\{\mu_n\}$, $n\in\Lambda'$, to $\mu$ it follows that the $\phi_1$-normalized potentials of the signed measures $\mu_n$, $n\in\Lambda'$, converge in $L^p(\widehat{\mathbb{C}})$ to the $\phi_1$-normalized potential of $\mu$, that is, $(\widehat{\mu_n})_{\phi_1}\to (\widehat{\mu})_{\phi_1}$ in $L^p(\widehat{\mathbb{C}})$ as $n\to\infty$, $n\in\Lambda'$. Moreover, by the definition of $\mu_n$ and by virtue of the normalization (8) of the polynomials $D_n^*$, for all $n\in\mathbb{N}$ we have $(\widehat{\mu_n})_{\phi_1}=((2m-2)n)^{-1}\log |D_n^*|$. Therefore, it remains to show that $(\widehat{\mu})_{\phi_1}=u_m$. Since we have proved that $((2m-2)n)^{-1}\log|D_n|\to u_m$ in $L^p_{\mathrm{loc}}(\widehat{\mathbb{C}}\setminus F_m)$ as $n\to\infty$ and $((2m-2)n)^{-1}\log |D_n^*|\to (\widehat{\mu})_{\phi_1}$ in $L^p(\widehat{\mathbb{C}})$ as $n\to\infty$, $n\in\Lambda'$, it follows from the relation $D_n^*=d_nD_n$ (where the $d_n$ are constants) that, as $n\to\infty$, $n\in\Lambda'$,
$$
\begin{equation*}
\frac1{(2m-2)n}\log d_n\to (\widehat{\mu})_{\phi_1}-u_m \quad\text{in } L^p_{\operatorname{loc}}(\widehat{\mathbb C}\setminus F_m).
\end{equation*}
\notag
$$
Since the $d_n$ are constants, we have $(\widehat{\mu})_{\phi_1}-u_m=\mathrm{const}$ in $L^p(\widehat{\mathbb{C}})$. From the normalizations (10) of $u_m$ and (33) of the potential $(\widehat{\mu})_{\phi_1}$ it follows that $\mathrm{const}=0$. Hence $(\widehat{\mu})_{\phi_1}=u_m$. Parts (2) and (3) of the Theorem 2 are proved. 4.3. Proofs of parts (4) and (5) of Theorem 2Parts (4) and (5) of Theorem 2 will easily be deduced from Statement 2 proved below. Let $a_1,a_2,\dots,a_{J}$ be the projections of all the zeros and poles (ignoring multiplicities) of the functions $A_j(\mathbf{z})$, $j=0,\dots,m$, in $\pi^{-1}(\widehat{\mathbb{C}}\setminus F_m)$. Also let $w_1,w_2,\dots,w_W$ be all zeros and poles of the function $\Pi_m(z)$ in $\mathbb{C}\setminus F_m$ (ignoring multiplicities). Recall that the points $p_k=\pi(\mathbf{p}_k)$ and $q_k(n)=\pi(\mathbf{q}_k(n))$, $k=1,\dots,M$, are the projections of the poles and the zeros of the function $R_n(\mathbf{z})$, respectively. For any $\varepsilon>0$ and any point $z^*\in\widehat{\mathbb{C}}$ let $O_{z^*}^{\varepsilon}$ denote the disc of radius $\varepsilon$ with centre $z^*$ in the spherical metric:
$$
\begin{equation*}
O_{z^*}^{\varepsilon}:=\{z\in\widehat{\mathbb C}\colon \operatorname{dist}(z,z^*)<\varepsilon\}.
\end{equation*}
\notag
$$
For any compact set $K\subset\widehat{\mathbb{C}}$ and any $\varepsilon>0$ set
$$
\begin{equation*}
K^{\varepsilon}(n):=K\setminus\biggl(\bigcup_{k=1}^{J} O_{a_k}^{\varepsilon}\cup \bigcup_{k=1}^W O_{w_k}^{\varepsilon}\cup \bigcup_{k=1}^M O_{p_k}^{\varepsilon}\cup \bigcup_{k=1}^M O_{q_k(n)}^{\varepsilon} \biggr).
\end{equation*}
\notag
$$
Statement 2. For any compact set $K\subset\widehat{\mathbb{C}}\setminus F_m$ and any $\varepsilon>0$ and, moreover, Proof. From representations (25) for $D_n$ and (23) for $Q_{n,m}$ it follows that for ${z\in\widehat{\mathbb{C}}\setminus F_m}$ (in particular, for $z\in K$)
$$
\begin{equation}
\begin{aligned} \, \notag \frac{D_n(z)}{Q_{n,m}^{2m-2}(z)} &= \frac{R_n^{2m-2}(\mathbf z^{(m)})A_m^{2m-2}(\mathbf z^{(m)})\Pi_m(z)(1+H_n(z))} {R_n^{2m-2}(\mathbf z^{(m)})A_m^{2m-2}(\mathbf z^{(m)})(1+h_{n,m}(z))^{2m-2}} \\ &=\Pi_m(z)\frac{1+H_n(z)}{(1+h_{n,m}(z))^{2m-2}}. \end{aligned}
\end{equation}
\tag{59}
$$
In the proof of Statement 4 in [24] (see [24], formula (87)) it was shown that
$$
\begin{equation}
\lim_{n\to\infty}\max_{z\in K^{\varepsilon}(n)}|h_{n,j}(z)|=0,\qquad j=0,\dots,m.
\end{equation}
\tag{60}
$$
Set
$$
\begin{equation*}
K_1^{\varepsilon}:=K\setminus\biggl(\bigcup_{k=1}^{J} O_{a_k}^{\varepsilon}\cup \bigcup_{k=1}^W O_{w_k}^{\varepsilon}\biggr).
\end{equation*}
\notag
$$
The meromorphic functions $A_j(\mathbf{z}^{(m)})$, $j=0,\dots,m$, and $\Pi_m(z)$ have neither zeros nor poles in the compact set $K_1^{\varepsilon}$, and so the absolute value of each of them is bounded and separated from zero there. Consequently, the absolute values of all the coefficients $M_{j_1,j_2,\dots,j_m}(z)$ in the representation (26) for $H_n(z)$ are also bounded on $K_1^{\varepsilon}$. Therefore, using that $K^\varepsilon(n)\subset K_1^{\varepsilon}$, from (26) and (60) we conclude that
$$
\begin{equation}
\lim_{n\to\infty}\max_{z\in K^{\varepsilon}(n)}|H_n(z)|=0.
\end{equation}
\tag{61}
$$
Bearing in mind that the function $\Pi_m(z)$ is bounded on $K_1^\varepsilon\supset K^\varepsilon(n)$ and taking (60) and (61) into account, from representation (59) we obtain (57).
Now let us prove (58). From (59) we obtain
$$
\begin{equation*}
\begin{aligned} \, \frac{D_n(z)}{Q_{n,m}^{2m-2}(z)}-\Pi_m(z) &=\Pi_m(z)\biggl(\frac{1+H_n(z)}{(1+h_{n.m}(z))^{2m-2}}-1\biggr) \\ &=\Pi_m(z)\frac{H_n(z)-h_{n,m}(z)B_n(z)}{(1+h_{n,m}(z))^{2m-2}}, \end{aligned}
\end{equation*}
\notag
$$
where $B_n(z):=\sum_{k=1}^{2m-2}\binom{{2m-2}}{k} (h_{n,m}(z))^{k-1}$. Then
$$
\begin{equation}
\biggl|\frac{D_n(z)}{Q_{n,m}^{2m-2}(z)}-\Pi_m(z)\biggr|\leqslant |\Pi_m(z)|\frac{|H_n(z)|+|h_{n,m}(z)||B_n(z)|}{|1+h_{n,m}(z)|^{2m-2}}.
\end{equation}
\tag{62}
$$
In the proof of Statement 4 in [24] (see [24], formula (90)) it was shown that for all $j=0,\dots,m$, for some constants $C_j>0$
$$
\begin{equation}
|h_{n,j}(z)|\leqslant C_j\exp(-n(u_m(z)-u_{m-1}(z))) \quad\text{as } z\in K^{\varepsilon}(n).
\end{equation}
\tag{63}
$$
Again, taking into account that the absolute values of the functions $A_j(\mathbf{z}^{(m)})$, $j=0,\dots,m$, and $\Pi_m(z)$ are separated from zero on $K_1^\varepsilon\supset K^\varepsilon(n)$ and that the $M_{j_1,j_2,\dots,j_m}(z)$ are bounded on $K_1^\varepsilon\supset K^\varepsilon(n)$, from (26) and (63) we obtain a similar inequality for $H_n(z)$: there exists a positive constant $\widetilde C$ such that
$$
\begin{equation}
|H_n(z)|\leqslant \widetilde C\exp(-n(u_m(z)-u_{m-1}(z))) \quad\text{as } z\in K^{\varepsilon}(n).
\end{equation}
\tag{64}
$$
Based on (60), one can find $N\in\mathbb{N}$ such that for $n>N$ and $z\in K^{\varepsilon}(n)$ the inequalities $|B_n(z)|\leqslant 1$ and $|1+h_{n,m}(z)|>1/2$ hold. Taking into account the boundedness of $\Pi_m(z)$ on $K_1^\varepsilon\supset K^\varepsilon(n)$ and estimates (63) and (64), from (62) we obtain (58). Statement 2 is proved. Let us deduce parts (4) and (5) of Theorem 2 from Statement 2. Let $K$ be a compact set in $\mathbb{C}\setminus F_m$. Let
$$
\begin{equation*}
r:=\max_{\zeta\in K}\operatorname{dist}(0;\zeta)+\frac1 3 \operatorname{dist}(K,\infty) \quad\text{and}\quad R:=\max_{\zeta\in K}\operatorname{dist}(0;\zeta)+\frac2 3 \operatorname{dist}(K,\infty).
\end{equation*}
\notag
$$
Then for $\varepsilon<\frac1 3 \operatorname{dist}(K,\infty)$, only those discs of radius $\varepsilon$ in the spherical metric can intersect $K$ whose centres belong to $\overline{O_0^r}$. All such discs belong to $\overline{O_0^R}$. Since on $\overline{O_0^R}$ the spherical metric is majorized by the Euclidean metric with some constant $C$, we see that the discs of radius $\varepsilon$ in the spherical metric with centres at those points $a_k$, $w_k$, $p_k$ and $q_k(n)$ that belong to $\overline{O_0^r}$ are subsets of the discs of radius $C\varepsilon$ in the Euclidean metric with the same centres. The number of such discs does not exceed $J+W+2M$, so from the standard estimate for the capacity of a union of sets (see, for instance, [36], Theorem 5.1.4) we obtain that the capacity of the union of these discs does not exceed $\mathrm{const}(C\varepsilon)^{1/(J+W+2M)}$. Thus, $\mathrm{cap}(K\setminus K^\varepsilon(n))\leqslant \mathrm{const}(C\varepsilon)^{1/(J+W+2M)}$ and we obtain parts (4) and (5) of Theorem 2 from Statement 2.
§ 5. The proofs of Theorems 3 and 45.1. The proof of Theorem 3Recall that $a\in\mathbb{C}\setminus F_m$ is a point such that $\Pi_m(a)=0$. Set Then $E_n$ is a meromorphic function in $\widehat{\mathbb{C}}\setminus F_m$. The idea of the proof of Theorem 3 is to find $c>0$ and $0<A<1$ such that for each sufficiently large $n\in\mathbb{N}$ one can find a radius $r(n)$ with the following properties: $0<r(n)<cA^n$, the function $\Pi_m$ has no poles in the closed Euclidean disc $\overline{B_a^{r(n)}}$ and $|E_n(z)|<|\Pi_m(z)|$ as $|z-a|=r(n)$. Then, applying Rouché’s theorem to the functions $\Pi_m(z)$ and $E_n(z)$ in the disc $B_a^{r(n)}$ we obtain that the function $D_n/Q_{n,m}^{2m-2}$, and therefore also $D_n$, has at least one zero in $B_a^{r(n)}$.From (59) we obtain
$$
\begin{equation}
E_n(z)= \Pi_m(z)\frac{\widetilde E_n(z)}{(1+h_{n,m}(z))^{2m-2}},
\end{equation}
\tag{65}
$$
where $\widetilde E_n(z):=H_n(z)+1-(1+h_{n,m}(z))^{2m-2}$. By (26)
$$
\begin{equation}
\widetilde E_n(z)=\sum_{\substack{0\leqslant j_0,j_1,\dots, j_m\leqslant 2m-2\\ 1\leqslant j_0+j_1+\dots+j_m\leqslant 2m-2}} \widetilde M_{j_1,j_2,\dots,j_m}(z) h_{n,0}^{j_0}(z)\dotsb h_{n,m}^{j_m}(z),
\end{equation}
\tag{66}
$$
where
$$
\begin{equation}
\widetilde M_{j_1,j_2,\dots,j_m}(z):= \begin{cases} \displaystyle \dfrac{M_{j_1,j_2,\dots,j_m}(z)}{A_m(\mathbf z^{(m)})^{2m-2}\Pi_m(z)}- \binom{2m-2}{j_m} &\text{for } j_0=\dots=j_{m-1}=0; \\ \dfrac{M_{j_1,j_2,\dots,j_m}(z)}{A_m(\mathbf z^{(m)})^{2m-2}\Pi_m(z)} &\text{otherwise}. \end{cases}
\end{equation}
\tag{67}
$$
Let $\delta_0>0$ be such that all functions $A_j(\mathbf{z}^{(m)})$ and $\Pi_m(z)$ have neither zeros nor poles in $\overline{B_a^{\delta_0}}\setminus a$. Then by Proposition 1 all coefficients $\widetilde M_{j_1,j_2,\dots,j_m}$ from (67) are meromorphic functions in $\widehat{\mathbb{C}}\setminus F_m$ and have no poles in $\overline{B_a^{\delta_0}}\setminus a$. Let $\Theta_1$ be the maximum order of poles of $\widetilde M_{j_1,j_2,\dots,j_m}$ at $a$. Then there exists a positive constant $C_0$ such that in $\overline{B_a^{\delta_0}}$, for all $j_1,j_2,\dots,j_m$ we have
$$
\begin{equation}
|\widetilde M_{j_1,j_2,\dots,j_m}(z)|\leqslant C_0|z-a|^{-\Theta_1}.
\end{equation}
\tag{68}
$$
Applying Lemma 1 to $z_0=a$, we obtain that there exist $\delta(a)\in(0;1)$ and $C(a)>0$ such that for $z\in B_a^{\delta(a)}$ the estimate
$$
\begin{equation}
|h_{n,j}(z)|\leqslant \frac{C(a) \exp(-n(u_m(z)-u_{m-1}(z)))}{|z-a|^{\Theta} \prod_{k=1}^{M} \min\{|z-q_k(n)|,\delta(a)\}}
\end{equation}
\tag{69}
$$
holds. Set $\delta=\min\{\delta_0,\delta(a)\}$ and
$$
\begin{equation*}
\varkappa:=\min_{\overline{B_a^{\delta}}}(u_m(z)-u_{m-1}(z)).
\end{equation*}
\notag
$$
Then $\varkappa>0$, since $u_m(z)>u_{m-1}(z)$ in $\mathbb{C}\setminus F_m$ by definition, and it was shown in [24], Lemma 1, that all functions $u_j$ are continuous in $\mathbb{C}$. From (69) we obtain that for $z\in B_a^\delta$ where
$$
\begin{equation}
e_n(z):= \frac{C(a) e^{-n\varkappa}}{|z-a|^{\Theta} \prod_{k=1}^{M} \min\{|z-q_k(n)|,\delta\}}.
\end{equation}
\tag{71}
$$
From (66), (68) and (70) it follows that for $z\in B_a^{\delta}$
$$
\begin{equation}
|\widetilde E_n(z)|\leqslant (2m-1)^{m+1}C_0|z-a|^{-\Theta_1}\max\{e_n(z),e_n^2(z),\dots,e_n^{2m-2}(z)\}.
\end{equation}
\tag{72}
$$
Now let $r>0$ be a positive number such that $(M+1)r<\delta$. Among the $M+1$ circles $\{z\in\mathbb{C}\colon |z-a|=lr\}$ lying in $B_a^\delta$, $l=1$,…, $M+1$, there exists a circle $S_n(r)$ such that the distances to it of all points $q_k(n)$, $k=1,\dots, M$, are at least $r/2$. Then for $e_n(z)$ from (71) the estimate is true on the circle $S_n(r)$.From representation (65) for the function $E_n(z)$ it is obvious that in order that $|E_n(z)|<|\Pi_m(z)|$ on the circle $S_n(r)$, it is sufficient that $|e_n(z)|<1/2$ and $|\widetilde E_n(z)|<2^{2-2m}$ there. By (73), in order that $|e_n(z)|<1/2$ on $S_n(r)$, it is sufficient that which is equivalent to the condition
$$
\begin{equation}
r>C_1\exp\biggl(-\frac {n\varkappa}{\Theta+M}\biggr),
\end{equation}
\tag{74}
$$
where $C_1:=(2^{M+1}C(a))^{1/(\Theta+M)}$. Let $r>0$ satisfy (74). Then, since $|e_n(z)|<1/2$ on the circle $S_n(r)$, the following estimate is true there by (72):
$$
\begin{equation}
|\widetilde E_n(z)|\leqslant \frac{C_0}{2} (2m-1)^{m+1}r^{-\Theta_1} e^{-n\varkappa}.
\end{equation}
\tag{75}
$$
Therefore, for $|\widetilde E_n(z)|<2^{2-2m}$ to hold on $S_n(r)$, it is sufficient that the conditions (74) and
$$
\begin{equation}
\frac{C_0}{2} (2m-1)^{m+1}r^{-\Theta_1}e^{-n\varkappa}< 2^{2-2m}
\end{equation}
\tag{76}
$$
hold. If $\Theta_1>0$, then (76) is true if and only if where $C_2:=\bigl(2^{2m-3} (2m-1)^{m+1}C_0\bigr)^{1/\Theta_1}$. If $\Theta_1=0$, then (76) holds true independently of $r$ for $n>N_1$, where $N_1\in\mathbb{N}$ is some constant. In this case let $C_2:=0$. Set $c:=(M+1)\max\{C_1,C_2\}+1$ and $A:=\exp\bigl(-\varkappa/(\max\{\Theta+M,\Theta_1\}\bigr)$. Let $r_n:=cA^n/(M+1)$ for $n\in\mathbb{N}$. Since $A<1$, there exists $N_0\in\mathbb{N}$ such that $(M+1)r_n<\delta$ for all $n>N_0$. For $\Theta_1=0$ we choose $N_0\geqslant N_1$. For ${n>N_0}$ and $r= r_n$ inequalities (74) and (76) hold, and this means that $|e_n(z)|<1/2$ and $|\widetilde E_n(z)|<2^{2-2m}$, and, consequently, also $|E_n(z)|<|\Pi_m(z)|$ on $S_n(r_n)$. Let $r(n)$ denote the radius of the circle $S_n(r_n)$. Applying Rouché’s theorem on the disc $B_a^{r(n)}$ to the functions $\Pi_m(z)$ and $E_n(z)$, we obtain that the difference between the number of zeros and the number of poles (taking account of multiplicities) of the function $\Pi_m(z)+E_n(z)=D_n(z)/Q_{n,m}^{2m-2}(z)$ in $B_a^{r(n)}$ is equal to the number of zeros of $\Pi_m(z)$ in the same disc, and therefore it is at least $1$. Consequently, there is a zero $z_n$ of $D_n(z)/Q_{n,m}^{2m-2}(z)$, that is, a zero of $D_n(z)$ in $B_a^{r(n)}$. Then $|z_n-a|<r(n)\leqslant(M+1)r_n=cA^n$. Theorem 3 is proved. 5.2. The proof of Theorem 4First we show that under the assumptions of Theorem 4 the function $\Pi_m(z)$ has either a zero of odd order or a pole of odd order at $a$. Let $\delta>0$ be such that $\pi$ has no critical points in $B_a^\delta\setminus a$ and $f$ has no poles in $\pi^{-1}(B_a^\delta\setminus a)$. Let $\mathbf{a}$ be the unique critical point of $\pi$ in $\pi^{-1}(a)$ (by the assumptions of the theorem it is of second order). Let $\mathbf{O}$ be the connected component of $\pi^{-1}(B_a^\delta)$ that contains $\mathbf{a}$. Let us choose a holomorphic coordinate $\zeta$ in $\mathbf{O}$ such that $z=a+\zeta^2(\mathbf{z})$ in $\mathbf{O}$. For each $z\in B_a^\delta\setminus a$ let $\pi^{-1}(z)\cap \mathbf{O}=\{\mathbf{z}^*,\mathbf{z}^{**}\}$ and $\mathbf{a}^*=\mathbf{a}^{**}=\mathbf{a}$. Then $\zeta(\mathbf{z}^{**})=-\zeta(\mathbf{z}^*)$ in $\mathbf{O}$ and the function $f$ is represented as where $f_1(z)$ and $f_2(z)$ are meromorphic in $B_a^\delta$ and are found by the formulae
$$
\begin{equation}
f_1(z)=\frac 1 2(f(\mathbf z^{*})+f(\mathbf z^{**}))\quad\text{and}\quad f_2(z)=\frac 1 {2\zeta(\mathbf z^*)}(f(\mathbf z^*)-f(\mathbf z^{**})).
\end{equation}
\tag{77}
$$
Note that $f_2(z)\not\equiv0$, since otherwise $\mathbf{a}$ is not a critical point of the projection $\pi$. By virtue of Remark 1 in $B_a^\delta\setminus a$ the representation
$$
\begin{equation}
\Pi_m(z)= (-1)^{m(m-1)/2}\prod_{\substack{\mathbf z_1,\mathbf z_2\in\pi^{-1}(z)\setminus \mathbf z^{(m)}\\ \mathbf z_1\ne\mathbf z_2}} (f(\mathbf z_1)-f(\mathbf z_2))
\end{equation}
\tag{78}
$$
holds. Since the projection $\pi$ is bijective on all the connected components of $\pi^{-1}(B_a^\delta)$ except $\mathbf{O}$, the product of all factors in (78) that do not contain $f(\mathbf{z}^*)$ and $f(\mathbf{z}^{**})$ is the square of a meromorphic function in $B_a^\delta$ (because each factor occurs twice with opposite signs). If $\mathbf{z}\in\pi^{-1}(z) \setminus\{\mathbf{z}^{(m)}; \mathbf{z}^*; \mathbf{z}^{**}\}$, then, using (77) and the fact that $\zeta^2(\mathbf{z}^{*})=z-a$, we obtain
$$
\begin{equation*}
\begin{aligned} \, &(f(\mathbf z)-f(\mathbf z^*))(f(\mathbf z)-f(\mathbf z^{**}))=f^2(\mathbf z)-f(\mathbf z)(f(\mathbf z^*)+f(\mathbf z^{**}))+f(\mathbf z^*)f(\mathbf z^{**}) \\ &\qquad =f^2(\mathbf z)-2f(\mathbf z)f_1(z)+(f_1^2(z)-(z-a)f_2^2(z)) \end{aligned}
\end{equation*}
\notag
$$
is a meromorphic function in $B_a^\delta$. Since each such product occurs twice in (78), the product of all factors in (78) that contain precisely one of the expressions $f(\mathbf{z}^*)$ and $f(\mathbf{z}^{**})$ is also the square of a meromorphic function in $B_a^\delta$. The remaining product in (78), namely, $(f(\mathbf{z}^*)-f(\mathbf{z}^{**}))(f(\mathbf{z}^{**}) -f(\mathbf{z}^{*}))= -\zeta^2(\mathbf{z}^*)f_2^2(z)=-(z-a)f_2^2(z)$, is a meromorphic function in $B_a^\delta$ with zero or pole of odd order at $a$. Therefore, the order of the zero or pole of the whole product (78) at the point $a$ is odd. To prove Theorem 4 we note that all the estimates obtained in the proof of Theorem 3 remain valid in our case. Thus, there exist constants $c>0$ and ${0<A<1}$ such that for all sufficiently large $n$, $n>N$, there exists a radius $r(n)<cA^n$ such that there are neither zeros nor poles of $\Pi_m(z)$ in $\overline{B_a^{r(n)}}\setminus a$ and for the function $E_n$ (see § 5.1) on the circle $\{z\in\mathbb{C}\colon |z-a|=r(n)\}$ the inequality $|E_n(z)|<|\Pi_m(z)|$ holds. Applying Rouché’s theorem again we obtain that the difference between the numbers of zeros and poles (taking account of multiplicities) of the function $\Pi_m(z)+E_n(z)={D_n(z)}/{Q_{n,m}^{2m-2}(z)}$ in the disc $B_a^{r(n)}$ is equal to the difference between the numbers of zeros and poles (taking account of multiplicities) of $\Pi_m(z)$ in $B_a^{r(n)}$ (that is, the order of zero or minus the order of the pole of $\Pi_m$ at the point $a$), and therefore, as follows from what we proved above, it is odd. Since each zero of the polynomial $Q_{n,m}$ that falls in $B_a^{r(n)}$ makes an even contribution (which is, moreover, a multiple of $2m-2$) to this difference, we see that the discriminant $D_n$ must have a zero $z_n$ in the disc $B_a^{r(n)}$. Theorem 4 is proved.
Citation in format AMSBIB
\Bibitem{KomPal24} |
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