On polynomials orthogonal to all powers of a given polynomial on a segment

F. Pakovich

Department of Mathematics, Ben Gurion University P.O.B. 653, Beer Sheva 84105, Israel e-mail: pakovich@math.bgu.ac.il

1 Introduction

In this paper we investigate the following “polynomial moment problem”: for a complex polynomial P (z) and distinct complex numbers a, b to describe polynomials q(z) such that

∫_{a}^{b} P^{i}(z)q(z)dz = 0  (1)

for all integer non-negative i. Despite its rather classical setting this problem attracted attention only recently in the series of papers [1]-[8], [23], where (1) arose in connection with the center problem for the Abel differential equation with polynomial coefficients in the complex domain. Posed initially as an intermediate step toward the center problem, the polynomial moment problem turned out to be quite delicate question unexpectedly involving such branches of mathematics as combinatorics and Galois theory.

For the simplest example P (z) = z the answer follows from the Stone-Weierstrass theorem: since the only continuous complex-valued function which is orthogonal to all powers of z on a segment is zero, the only polynomial solution to (1) is q(z) = 0. On the other hand, for P (z) = z^{2} and [a, b] = [−1, 1] non-trivial polynomial solutions to (1) already exist since any polynomial q(z) such that q(−z) = −q(z) is clearly satisfied (1). Actually, for any P (z) ∈ ℂ[z], a, b ∈ ℂ such that P (a) = P (b), non-trivial polynomial solutions to (1) exist. Indeed, it is enough to set q(z) = R′(P (z))P′(z), where R(z) is any complex polynomial. Then for any i ≥ 0 we have:

∫_{a}^{b} P^{i}(z)q(z)dz = ∫_{P (a)}^{P (b)} y^{i}R′(y)dy = 0.

More generally, the following “composition condition” imposed on P (z) and Q(z) = ∫ q(z)dz is sufficient for polynomials P (z), q(z) to satisfy (1): there exist polynomials P̃(z), Q̃(z), W (z) such that

P (z) = P̃(W (z)), Q(z) = Q̃(W (z)), and W (a) = W (b).  (2)

The sufficiency of condition (2) follows from W (a) = W (b) after the change of variable z → W (z). It was suggested in the papers [2]-[6] (“the composition conjecture”) that, under the additional assumption P (a) = P (b), condition (1) is actually equivalent to condition (2). This conjecture is shown to be true if the collection P (z), a, b is generic enough. For instance, if a, b are not critical points of P (z) ([9]) or if P (z) is indecomposable ([15]) (see also [18], [20], and the papers cited above). Nevertheless, in general the composition conjecture fails to be true.

A class of counterexamples to the composition conjecture was constructed in [14]. These counterexamples exploit polynomials which admit “double decompositions” of the form P (z) = A(B(z)) = C(D(z)), where A(z), B(z), C(z), D(z) are non-linear polynomials. If P (z) is such a polynomial and, in addition, the equalities B(a) = B(b), D(a) = D(b) hold, then for any polynomial Q(z), which can be represented as Q(z) = E(B(z)) + F (D(z)) for some polynomials E(z), F (z), condition (1) is satisfied with q(z) = Q′(z) by linearity. On the other hand, it can be shown (see [14]) that if deg B(z) and deg D(z) are coprime then condition (2) is not satisfied already for Q(z) = B(z) + D(z). Note that, by the second Ritt theorem, double decompositions with deg A(z) = deg D(z), deg B(z) = deg C(z) and deg B(z), deg D(z) coprime are equivalent either to decompositions with

A(z) = z^{n}R^{m}(z), B(z) = z^{m}, C(z) = z^{m}, D(z) = z^{n}R(z^{m})

where R(z) is a polynomial and GCD(n, m) = 1, or to decompositions with

A(z) = T_{m}(z), B(z) = T_{n}(z), C(z) = T_{n}(z), D(z) = T_{m}(z),

where T_{n}(z), T_{m}(z) are Chebyshev polynomials and GCD(n, m) = 1 (see [19], [21]). In particular, the simplest explicit counterexample to the composition conjecture has the following form:

P (z) = T_{6}(z), q(z) = T′_{2}(z) + T′_{3}(z), a = −\sqrt{3}/2. b = \sqrt{3}/2.

The counterexamples above suggest to transform the composition conjecture as follows ([17]): non-zero polynomials P (z), q(z) satisfy condition (1) if and only if Q(z) = ∫ q(z)dz can be represented as a sum of polynomials Q_{j} such that

P (z) = P̃_{j}(W_{j}(z)), Q_{j}(z) = Q̃_{j}(W_{j}(z)), and W_{j}(a) = W_{j}(b) (3)

for some P̃_{j}(z), Q̃_{j}(z), W_{j}(z) ∈ ℂ[z]. Note that we do not make any additional assumptions about the values of P (z) at the points a, b any more. In particular, the conjecture implies that non-zero polynomials orthogonal to all powers of a given polynomial P (z) on [a, b] exist if and only if P (a) = P (b). Note also that while condition (2) via Lüroth’s theorem essentially reduces to the requirement that the field ℂ(P, Q) is a proper subfield of ℂ(z) or equivalently to equality (5) below, condition (3) does not have a similar interpretation. For the case P (z) = T_{n}(z) conjecture (3) was proved in [16].

This paper is organized as follows. In the second section we prove the main result of this paper: a criterion for a polynomial q(z) to satisfy (1) given in terms of linear relations between branches of the algebraic function Q(P^{−1}(z)), where P^{−1}(z) denotes the algebraic function which is inverse to P (z). More precisely, let P_{i}^{−1}(z), 1 ≤ i ≤ n, be single-valued branches of P^{−1}(z) in a simply-connected domain U ⊂ ℂ containing no critical values of P (z). We show that there exists a system of equations

∑_{i=1}^{n} f_{s,i}Q(P_{i}^{−1}(z)) = 0,  1 ≤ s ≤ deg P (z),  (4)

depending on P (z), a, b, with f_{s,i} taking values in the set {0, −1, 1}, such that (1) holds if and only if (4) holds with Q(z) = ∫ q(z)dz. Furthermore, we give a method for finding (4) explicitly via a special graph λ_{P} embedded into the Riemann sphere, called the “cactus” of P (z). The graph λ_{P} , like similar objects named “S-graphs”, “pictures”, or “dessins d’enfants”, provides a full combinatorial description of the monodoromy of P (z), and, in particular, permits to relate properties of the collection P (z), a, b which are connected with the polynomial moment problem to combinatorial properties of the pair consisting of the tree λ_{P} and the path Γ_{a,b} connecting points a, b on λ_{P} .

The criterion (4) has a number of applications. For example, it allows us to reduce an infinite set of equations (1) to a finite set of equations w_{k} = 0, 0 ≤ k ≤ M, where w_{k} are initial coefficients of the Puiseux expansions of the combinations of branches in (4) and M depends only on degrees of P (z) and Q(z). Furthermore, using the equivalence of (1) and (4), one can provide a variety of different conditions on a collection P (z), a, b under which (1) and (2) are equivalent - this is the subject of the third section of this paper. Essentially the finding of such conditions, which are of interest because of applications to the Abel equation (see [1], [7], [8]), reduces to the finding conditions under which system (4) implies that

Q(P_{i_{1}}^{−1}(z)) = Q(P_{i_{2}}^{−1}(z))  (5)

for some i_{1} ≠ i_{2}. In its turn these conditions can be naturally given in terms of combinatorics of the graph λ_{P} . Finally, note that criterion (4) permits to use in the study of the polynomial moment problem the methods of Galois theory since system (4) can be interpreted as a system of relations between roots of the corresponding irreducible polynomial which defines the algebraic function Q(P^{−1}(z)) (see e. g. subsection 5.3 below).

While an explicit form of system (4) depends on the collection P (z), a, b, there exists a necessary condition for (1) to be satisfied the form of which is invariant with respect to P (z), a, b. Namely, it is known ([15], [18]), and also can be deduced directly from (4), that (1) implies the equality

\frac{1}{d_{a}} ∑_{s=1}^{d_{a}} Q(P_{a_{s}}^{−1}(z)) = \frac{1}{d_{b}} ∑_{s=1}^{d_{b}} Q(P_{b_{s}}^{−1}(z)),  (6)

if P (a) = P (b), or the system

\frac{1}{d_{a}} ∑_{s=1}^{d_{a}} Q(P_{a_{s}}^{−1}(z)) = 0,  \frac{1}{d_{b}} ∑_{s=1}^{d_{b}} Q(P_{b_{s}}^{−1}(z)) = 0,  (7)

if P (a) ≠ P (b), where P_{a_{1}}^{−1}(z), P_{a_{2}}^{−1}(z), ..., P_{a_{da}}^{−1} (z) (resp. P_{b_{1}}^{−1}(z), P_{b_{2}}^{−1}(z), ... , P_{b_{d_{b}}}^{−1}(z)) denote the branches of P^{−1}(z) which map points close to P (a) (resp. P (b)) to points close to a (resp. b). Note that if d_{a} = d_{b} = 1 that is if the points a, b are not critical points of P (z), then (6) reduces to (5) while (7) leads to the equality q(z) = 0.

In the fourth section of this paper we investigate equation (6) and system (7). In particular, we establish a specific geometric property of the monodromy groups of polynomials, related to the topology of the Riemann sphere, from which we deduce the following result: if P (z), Q(z) ∈ ℂ[z], deg P (z) = n, deg Q(z) = m satisfy (6) or (7), then for coefficients of the Puiseux expansions near infinity

Q(P_{i}^{−1}(z)) = ∑_{k=−m}^{∞} u_{k}ε_{n}^{ik}z^{− \frac{k}{n}}  (8)

the equality u_{k} = 0 holds whenever GCD(k, n) = 1. This fact agrees with conjecture (3) and, in particular, implies that for P (z), q(z) satisfying (1) the numbers n and m can not be coprime. As an application of the Puiseux expansions technique we show in the fifth section that conditions (1) and (2) are equivalent if at least one from points a, b is not a critical point of P (z) or if deg P (z) = p^{r} for a prime number p.

Finally, on the base of obtained results, in the sixth section we show that for any collection P (z), a, b with deg P (z) < 10 conditions (1) and (2) are equivalent except the case when P (z), a, b is linearly equivalent to the collection T_{6}(z), −\sqrt{3}/2, \sqrt{3}/2. Since for P (z) = T_{n}(z) all solutions to (1) were obtained in [16], this provides the complete solution of the polynomial moment problem for P (z), a, b with deg P (z) < 10.

Acknowledgments. I am grateful to M. Sodin and Y. Yomdin for interesting discussions and comments. Also I would like to thank the Max Planck Institut für Mathematik for its support and hospitality.

2 Criterion for a polynomial to be orthogonal to all powers of a given polynomial

2.1 Cauchy type integrals of algebraic functions

A quite general approach to the polynomial moment problem was proposed in the paper [18] concerning Cauchy type integrals of algebraic functions

I(t) = I(γ, g, t) = \frac{1}{2πi} ∫_{γ} \frac{g(z)dz}{z − t} .  (9)

In this subsection we briefly recall it (see [18] for details) and outline in this context the approach of this paper.

First of all notice that condition (1) is equivalent to the condition

∫_{a}^{b} P^{i}(z)Q(z)P′(z)dz = 0  (10)

for i ≥ 0, where Q(z) = ∫ q(z)dz is normalized by the condition Q(a) = Q(b) = 0 (Q(a) always equals Q(b) by (1) taken for i = 0). Furthermore, vice versa, condition (10) with Q(a) = Q(b) = 0 implies that (1) holds with q(z) = Q′(z). (Actually, it was the condition (10) that appeared initially in the papers on differential equations cited above).

Indeed, condition (10) is equivalent to the condition that the function

H(t) = ∫_{a}^{b} \frac{Q(z)P′(z)dz}{P (z) − t}

vanishes identically near infinity, since near infinity

H(t) = − ∑_{i=0}^{∞} m_{i}t^{−(i+1)}, where m_{i} = ∫_{a}^{b} P^{i}(z)Q(z)P′(z)dz.

On the other hand, we have:

\frac{dH(t)}{dt} = ∫_{a}^{b} \frac{Q(z)P′(z)dz}{(P (z) − t)^{2}} = − ∫_{a}^{b} Q(z)d (\frac{1}{P (z) − t}) =

= \frac{Q(a)}{P (a) − t} − \frac{Q(b)}{P (b) − t} + H̃(t), (11)

where

H̃(t) = ∫_{a}^{b} \frac{q(z)dz}{P (z) − t}.

Since near infinity

H̃(t) = − ∑_{i=0}^{∞} m̃_{i}t^{−(i+1)}, where m̃_{i} = ∫_{a}^{b} P^{i}(z)q(z)dz,

it follows from (11) that conditions (1) and (10) are equivalent whenever Q(a) = Q(b) = 0.

Furthermore, performing the change of variable z → P (z), we see that H(t) coincides with integral (9) where γ = P ([a, b]) and g(z) is an algebraic function obtained by the analytic continuation of a germ of the algebraic function g(z) = Q(P^{−1}(z)) along γ. Integral representation (9) defines a collection of univalent regular functions I_{i}(t); each I_{i}(t) is defined in a domain D_{i} of the complement of γ in ℂℙ^{1}. Denote by I_{∞}(t) the function defined in the domain D_{∞} containing infinity. Then the vanishing of H(t) near infinity becomes equivalent to the equality I_{∞}(t) ≡ 0.

More generally, consider integral (9), where γ is a curve in the complex plane ℂ and g(z) is any “piecewise-algebraic” function on γ. More precisely, we assume that after removing from γ a finite set of points Σ_{γ}, the set γ \ Σ_{γ} is a union of topological segments ∪γ_{s} such that for each γ_{s} there exists a domain V_{s} ⊃ γ_{s} and an analytic in V_{s} algebraic function g_{s}(z) such that g(z) on γ_{s} coincides with g_{s}(z). Furthermore, we assume that at the points of Σ_{γ}, the complete analytic continuations ĝ_{s}(z) of g_{s}(z) can ramify but do not have poles. Below we sketch conditions for I_{∞}(t) to be a rational function; if these conditions are satisfied, then in order to verify the equality I_{∞}(t) ≡ 0 it is enough to examine possible poles.

Denote by Σ_{g} the set of all singularities of ĝ_{s}(z) in ℂℙ^{1}. Show that any element (I_{i}(t), U_{i}) can be analytically continued along any curve S = S_{t_{1},t_{2}} connecting points t_{1}, t_{2} ∈ ℂℙ^{1} and avoiding points from the sets Σ_{g} and Σ_{γ}. First of all notice that if t_{2} ∈ ∂U_{i} then an analytical extension of (I_{i}(t), U_{i}) to a domain containing t_{2} is given simply by the integral I(γ̃, g, t), where γ̃ is a small deformation of γ such that t_{2} ∈ Ũ_{i}. Furthermore, if S = S_{t_{1},t_{2}} is a simple curve connecting points t_{1} ∈ U_{i}, t_{2} ∈ U_{j}, where U_{i}, U_{j} are domains with a common segment of the boundary γ_{s} and (g_{s}, V_{s}) is the corresponding algebraic function, then the well-known boundary property of Cauchy type integrals (see e.g. [13]) implies that

(I_{i}(t), U_{i} ∩ V_{s}) = (I_{j}(t), U_{j} ∩ V_{s}) + (g_{s}, V_{s}).

Therefore, the analytic continuation of (I_{i}(t), U_{i}) along S can be defined via the analytic continuation of the right side of this formula.

Finally, for arbitrary domains U_{i}, U_{j} and a curve S = S_{t_{1},t_{2}} connecting points t_{1} ∈ U_{i} and t_{2} ∈ U_{j}, the analytic continuation (I_{i}(t), U_{i})_{S} of the element (I_{i}(t), U_{i}) along S can be defined inductively as follows. Let S ∩ γ = {c_{1}, c_{2}, ..., c_{r}}, c_{s} ∈ V_{s}, 1 ≤ s ≤ r, and let (g_{1}, V_{1}), (g_{2}, V_{2}), ..., (g_{r}, V_{r}) be the corresponding algebraic functions. Define a germ g_{γ,S} of an algebraic function near the point t_{2} by the formula:

g_{γ,S} = ∑_{i=1}^{r}(g_{i}, V_{i})_{S_{c_{i}}} ,

where (g_{i}, V_{i})_{S_{c_{i}}}, 1 ≤ i ≤ r, denotes the analytic continuation of the element (g_{i}, V_{i}) (taken with the sign corresponding to the orientation of the crossing of S and γ) along a part of S from c_{i} to t_{2}. Then by induction we have:

(I_{i}(t), U_{i})_{S} = (I_{j}(t), U_{j}) + g_{γ,S}.  (12)

In particular, a complete analytic continuation Î_{i}(t) of the element (I_{i}(t), U_{i}) is a multi-valued analytic function with a finite set of singularities ΣÎ_{i} ⊂ Σ_{g} ∪ Σ_{γ}.

From formula (12) one deduces the following criterion ([18]): Î_{i}(t) is a rational function if and only if the equality

g_{γ,S} = 0  (13)

holds for any curve S = S_{t_{1},t_{2}} as above with t_{1} = t_{2} ∈ U_{i}. Indeed, the necessity of (13) is obvious. To establish the sufficiency observe that (13) implies, in particular, that Î_{i}(t) has no ramification in its singularities. Therefore, if z_{0} is a singularity of Î_{i}(t) such that z_{0} ∈ ℂℙ^{1} \ γ, then formula (12) implies that z_{0} is a pole the worst. On the other hand, if z_{0} ∈ Σ_{γ} then the function I_{j}(t) near z_{0} has the form

I_{j}(t) = u(t) log(t − z_{0}) + v(t),  (14)

where u(t) is a function analytic at z_{0} and v(t) is a bounded function which has a finite ramification at z_{0} (see [18]). Therefore, if Î_{i}(t) has no ramification at z_{0}, then necessarily u(t) ≡ 0 and hence z_{0} actually is a removable singularity of Î_{i}(t) since v(t) and ĝ_{s}(t) are bounded near z_{0}.

Although the method above in principle is constructive its practical application is rather difficult since the calculation of sums g_{γ,S} is complicated. In this paper we propose a modification of the method above designed specially for the polynomial moment problem. This modification permits to avoid any analytic continuations and allows us to obtain a necessary and sufficient conditions for equality (1) to be satisfied in a closed and convenient form. The idea is to choose a very special way of integration Γ connecting points a, b (we can use any of them since integrals (1) do not depend on Γ). It turns out that Γ can by chosen so that ℂℙ^{1} \ P (Γ) consists of a unique domain. Then condition I_{∞}(t) ≡ 0 simply reduces to the condition that the corresponding algebraic functions g_{s}(z) vanish on P (Γ). Furthermore, we choose Γ as a subset of a special tree λ_{P} embedded into the Riemann sphere, called the cactus of P (z), which contains all the information about the monodromy P (z). This allows us to relate properties of the collection P (z), a, b which are connected to the polynomial moment problem to combinatorics of the tree λ_{P} and the path connecting points a, b on λ_{P} .

2.2 Cacti

To visualize the monodromy group of a polynomial P (z) it is convenient to consider a graphical object λ_{P} called the cactus of P (z) (see e. g. [10], [12]).

Let c_{1}, c_{2}, ..., c_{k} be all finite critical values of P (z) and let c be a not critical value. Draw a star S joining c with c_{1}, c_{2}, ..., c_{k} by non intersecting arcs γ_{1}, γ_{2}, ..., γ_{k}. We will suppose that c_{1}, c_{2}, ..., c_{k} are numerated in such a way that in a counterclockwise rotation around c the arc γ_{s}, 1 ≤ s ≤ k − 1, is followed by the arc γ_{s+1}. By definition, the cactus λ_{P} is the preimage of S under the map P (z) : ℂ → ℂ. More precisely, we consider λ_{P} as a k + 1-colored graph embedded into the Riemann sphere: vertices of λ_{P} colored by the s-th color, where 1 ≤ s ≤ k, are preimages of the point c_{s}, vertices colored by the k + 1-th color (to be definite we will suppose that it is the white color) are preimages of the point c, and edges of λ_{P} are preimages of the arcs γ_{s}, 1 ≤ s ≤ k. It is not difficult to show that the graph λ_{P} is connected and has no cycles. Therefore, λ_{P} is a plane tree.

Figure 1

The valency of a non-white vertex z of λ_{P} coincides with the multiplicity of z with respect to P (z) while all white vertices of λ_{P} are of the same valency n = deg P (z). The set of all edges of λ_{P} adjacent to a white vertex w is called by a star of λ_{P} centered at w. Clearly, λ_{P} has nk edges and n stars. The set of stars of λ_{P} is naturally identified with the set of branches of P^{−1}(z) as follows. Let U be a simply connected domain containing no critical values of P (z) such that S \ {c_{1}, c_{2}, ..., c_{k}} ⊂ U. By the monodromy theorem in U there exist n single valued branches of P^{−1}(z). Any such a branch P_{i}^{−1}(z), 1 ≤ i ≤ n, maps S \ {c_{1}, c_{2}, ..., c_{k}} into a star of λ_{P} and we will label the corresponding star by the symbol S_{i} (see Fig. 1).

The cactus λ_{P} permits to reconstruct the monodromy group G_{P} of P (z). Indeed, G_{P} is generated by the permutations g_{s} ∈ S_{n}, 1 ≤ s ≤ k, where g_{s} is defined by the condition that the analytic continuation of the element (P_{i}^{−1}(z), U), 1 ≤ i ≤ n, along a counterclockwise oriented loop l_{s} around c_{s} is the element (P_{g_{s}(i)}^{−1}(z), U). Having in mind the identification of the set of stars of λ_{P} with the set of branches of P^{−1}(z), the permutation g_{s}, 1 ≤ s ≤ k, can be identified with the permutation ĝ_{s}, 1 ≤ s ≤ k, acting on the set of starts of λ_{P} in the following way: ĝ_{s} sends the star S_{i}, 1 ≤ i ≤ n, to the “next” one in a counterclockwise direction around its vertex of color s. For example, for the cactus shown on Fig. 1 we have: g_{1} = (1)(2)(37)(4)(5)(6)(8), g_{2} = (1)(2)(3)(47)(56)(8), g_{3} = (1238)(4)(57)(6).

Note that since P (z) is a polynomial, the permutation g_{∞} = g_{1}g_{2}...g_{k} is a cycle of length n. Usually, we will choose the numeration of S_{i}, 1 ≤ i ≤ n, in such a way that this cycle coincides with the cycle (12...n).

2.3 Criterion

In this subsection we give explicit necessary and sufficient conditions for P (z), q(z) ∈ ℂ[z] and a, b ∈ ℂ, a ≠ b, to satisfy (1), (10). For this propose we choose the way of integration Γ_{a,b} connecting a, b so that Γ_{a,b} would be a subset of λ_{P} .

More precisely, for any P (z) ∈ ℂ[z] and a, b ∈ ℂ let us define an extended cactus λ̃_{P} = λ̃_{P} (c_{1}, c_{2}, ... , ck̃) as follows. Let c_{1}, c_{2}, ..., ck̃ be all finite critical values of P (z) complemented by P (a) or P (b) (or by both of them) if P (a) or P (b) is not a critical value of P (z). Consider an extended star S̃ connecting c with c_{1}, c_{2}, ..., ck̃ and set λ̃_{P} = P^{−1}{S̃} (we suppose that c is chosen distinct from P (a), P (b)). Clearly, λ̃_{P} considered as a k̃ + 1 colored graph is still connected and has no cycles. Furthermore, by construction the points a, b are vertices of λ̃_{P} . Since λ̃_{P} is connected there exists an oriented path Γ_{a,b} ⊂ λ̃_{P} with the starting point a and the ending point b. Moreover, since λ̃_{P} has no cycles there exists exactly one such a path. We choose Γ_{a,b} as a new way of integration.

Let U be a domain as above and let Q(z) = ∫ q(z)dz be normalized by the condition Q(a) = Q(b) = 0. For each s, 1 ≤ s ≤ k̃, define a linear combination φ_{s}(z) of branches Q(P_{i}^{−1}(z)), 1 ≤ i ≤ n, in U as follows. Set

φ_{s}(z) = ∑_{i=1}^{n} f_{s,i}Q(P_{i}^{−1}(z)),  (15)

where f_{s,i} ≠ 0 if and only if the path Γ_{a,b} passes through a vertex v of the star S_{i} colored by the s-th color (we do not take into account the stars S_{i} for which Γ_{a,b} ∩ S_{i} contains only the point v). Furthermore, if under a moving along Γ_{a,b} the vertex v is followed by the center of S_{i} then f_{s,i} = −1 otherwise f_{s,i} = 1. As an example consider the cactus shown on Fig. 1. Then for the path Γ_{a,b} pictured by the fat line we have:

φ_{1}(z) = −Q(P_{2}^{−1}(z)) + Q(P_{3}^{−1}(z)) − Q(P_{7}^{−1}(z)),
φ_{2}(z) = Q(P_{7}^{−1}(z)) − Q(P_{4}^{−1}(z)),
φ_{3}(z) = Q(P_{2}^{−1}(z)) − Q(P_{3}^{−1}(z)) + Q(P_{4}^{−1}(z)).

Theorem 2.1 Let P (z), q(z) ∈ ℂ[z], a, b ∈ ℂ, a ≠ b, and let λ̃_{P} (c_{1}, c_{2}, ... , ck̃) be an extended cactus corresponding to the collection P (z), a, b. Then (1) holds if and only if the equality φ_{s}(z) ≡ 0 holds in U for any s, 1 ≤ s ≤ k̃.

Proof. Indeed, condition (10) is equivalent to the condition that the function

H(t) = ∫_{Γ_{a,b}} \frac{Q(z)P′(z)dz}{P (z) − t}

vanishes identically near infinity. On the other hand, using the change of variable z → P (z), we can express the function H(t) as a sum of Cauchy type integrals of algebraic functions as follows:

H(t) = ∑_{s=1}^{k̃} ∫_{γ_{s}} \frac{φ_{s}(z)}{z − t} dz.  (16)

Since this formula implies that H(t) is analytic in a domain V = ℂℙ^{1} \ S we see that the vanishing of H(t) near infinity is equivalent to the condition that H(t) ≡ 0 in V.

Let z_{0} be an interior point of γ_{s}, 1 ≤ s ≤ k̃. Then by the well-known boundary property of Cauchy type integrals (see e.g. [13]) we have:

lim_{t→z_{0}}^{+}H(t) − lim_{t→z_{0}}^{−}H(t) = φ_{s}(t_{0}),

where the limits are taken respectively for t tending to z_{0} from the “left” and from the “right” parts of V with respect to γ_{s}. If H(t) ≡ 0 in V, then

lim_{t→z_{0}}^{+}H(t) = lim_{t→z_{0}}^{−}H(t) = 0,

and, therefore, φ_{s}(z_{0}) = 0. Since this equality holds for any interior point z_{0} of any arc γ_{s}, 1 ≤ s ≤ k̃, we conclude that φ_{s}(z) ≡ 0, 1 ≤ s ≤ k̃, in U. On the other hand, if φ_{s}(z) ≡ 0, 1 ≤ s ≤ k̃, in U, then it follows directly from formula (16) that H(t) ≡ 0 in V.

Note that some of equations φ_{s}(z) ≡ 0, 1 ≤ s ≤ k̃, could be trivial. This happens exactly for the values s such that the path Γ_{a,b} does not pass through vertices colored by the s-th color. Note also that equations (15) are linearly dependent. Indeed, for each i such that there exists an index s, 1 ≤ s ≤ k̃, with f_{s,i} ≠ 0 there exist exactly two such indices s_{1}, s_{2} and c_{s_{1},i} = −c_{s_{2},i}. Therefore, the equality

∑_{s=1}^{k̃} φ_{s}(t) = 0

holds in U.

2.4 Checking the criterion

Let P (z), Q(z) ∈ ℂ[z], deg P (z) = n, deg Q(z) = m. Let U be a simply connected domain containing no critical values of P (z) and let P_{i}^{−1}(z), 1 ≤ i ≤ n, be branches of P^{−1}(z) in U. In this subsection we provide a simple estimation for the order of a zero in U of a function of the form

ψ(z) = ∑_{i=1}^{n} f_{i}Q(P_{i}^{−1}(z)), f_{i} ∈ ℂ,

via the degrees of P (z) and Q(z). This reduces the verification of the criterion to the calculation of a finite set of initial coefficients of Puiseux expansions of functions (15) and, as a corollary, provides a practical method for checking an infinite set of equation (1) in a finite number of steps.

Lemma 2.1 If ψ(z) ≠ 0 then ψ(z) satisfies an equation

y^{N}(z) + a_{1}(z)y^{N−1}(z) + ... + a_{N}(z) = 0,  (17)

where a_{j}(z) ∈ ℂ[z], a_{N}(z) ≠ 0, and N ≤ n!. Furthermore, deg a_{j}(z) ≤ (\frac{m}{n})^{j} , 1 ≤ j ≤ N.

Proof. Indeed, if ψ(z) ≠ 0 then, since ψ(z) is a sum of algebraic functions, ψ(z) itself is an algebraic function and therefore satisfies an algebraic equation (17) with a_{i}(z) ∈ ℂ(z), 1 ≤ j ≤ N. Furthermore, we can suppose that this equation is irreducible. Then a_{N}(z) ≠ 0 and the number N coincides with the number of different analytic continuations ψ_{j}(z) of ψ(z) along closed curves. Clearly, N can be bounded by the number N_{1} of different elements of the monodromy group of P (z). In its turn, N_{1} is bounded by the number of elements of the full symmetric group S_{n}. Hence, N ≤ n!.

Furthermore, since P (z), Q(z) are polynomials, the rational functions a_{j}(z), 1 ≤ j ≤ N, as the symmetric functions of ψ_{j}(z), 1 ≤ j ≤ N, have no poles in ℂ and therefore are polynomials. Finally, since near infinity branches P_{i}^{−1}(z), 1 ≤ i ≤ n, of P^{−1}(z) are represented by the Puiseux series

P_{i}^{−1}(z) = ∑_{k=−1}^{∞} v_{k}ε_{n}^{ik}z^{− \frac{k}{n}} ,  v_{k} ∈ ℂ,  ε_{n} = exp(2πi/n),  (18)

the first non-zero exponent of the Puiseux series at infinity for the functions ψ_{j}(z), 1 ≤ j ≤ N, is less or equal than m/n. It follows that deg a_{j}(z) ≤ (\frac{m}{n})^{j} , 1 ≤ j ≤ N.

Corollary 2.1 Let z_{0} ∈ U. To verify that ψ(z) ≡ 0 it is enough to check that the first (\frac{m}{n})^{n!} + 1 coefficients of the series ψ(z) = ∑_{k=0}^{∞} w_{k}(z − z_{0})^{k} vanish.

Proof. Indeed, suppose that ord_{z_{0}}ψ(z) > (\frac{m}{n})^{n!} but ψ(z) ≠ 0. Then, by lemma 2.1, ψ(z) satisfies (17), where deg a_{j}(z) ≤ (\frac{m}{n})^{j} ≤ (\frac{m}{n})^{n!} , 1 ≤ j ≤ N, and a_{N} ≠ 0. It follows that

ord_{z_{0}}{ψ^{N}(z)} > ord_{z_{0}}{a_{i_{1}}(z)ψ^{N−i_{1}}(z)} > ... > ord_{z_{0}}{a_{i_{k}}(z)ψ^{N−i_{k}}(z)},

where 0 < i_{1} < i_{2} < ... < i_{k} = N are all indices for which a_{i}(z) ≠ 0. Therefore,

ord_{z_{0}}{ψ^{N}(z) + a_{1}(z)ψ^{N−1}(z) + ... + a_{N}(z)} = ord_{z_{0}}{a_{N}(z)} < ∞

in contradiction with equality (17).

3 Definite polynomials

In this section, as a first application of theorem 2.1, we provide a number of conditions on a collection P (z), a, b, where P (z) ∈ ℂ[z], a, b ∈ ℂ, a ≠ b, under which conditions (1) and (2) are equivalent; such collections are called definite and are of interest because of applications to the Abel equation (see [1], [7], [8]).

3.1 A combinatorial condition for a change of variable

The simplest form of the equality φ_{s}(z) = 0 is equality (5). Furthermore, (5) has a clear compositional meaning.

Lemma 3.1 The equality (5) holds if and only if

P (z) = P̃(W (z)),  Q(z) = Q̃(W (z))  (19)

for some polynomials P̃(z), Q̃(z), W (z) with deg W (z) > 1.

The proof of this lemma easily follows from the Lüroth theorem (see e.g. [15], [20]). If condition (19) is satisfied we say that polynomials P (z), Q(z) have a (non-trivial) common right divisor in the composition algebra of polynomials.

Below we give a convenient combinatorial condition on a collection P (z), a, b which implies that for any q(z) satisfying (1) polynomials P (z), Q(z) = ∫ q(z)dz have a common right divisor in the composition algebra of polynomials. The use of this condition permits, after the change of variable z → W (z), to reduce the solution of the polynomial moment problem for a polynomial P (z) to that for a polynomial of lesser degree P̃(z).

Let λ̃_{P} be a k̃ + 1 colored extended cactus corresponding to a collection P (z), a, b and let Γ_{a,b} be the path connecting points a, b on λ̃_{P} . For each s, 1 ≤ s ≤ k̃, define the weight w(s) of the s-th color on Γ_{a,b} as a number of vertices v ∈ Γ_{a,b} colored by the s-th color with the convention that vertices a, b are counted with the coefficient 1/2. For example, for Γ_{a,b} shown on Fig. 1 we have w(1) = w(3) = 3/2, w(2) = 1.

Theorem 3.1 Let P (z), q(z) ∈ ℂ[z], q(z) ≠ 0, a, b ∈ ℂ, a ≠ b satisfy (1). Suppose that there exists s, 1 ≤ s ≤ k̃, such that w(s) = 1 on Γ_{a,b}. Then P (z), Q(z) have a common right divisor in the composition algebra.

Proof. Indeed, the construction of Γ_{a,b} implies that if w(s) = 1, then f_{s,i} ≠ 0 exactly for two values i_{1}, i_{2}, 1 ≤ i_{1}, i_{2} ≤ n. Moreover, for these values we have c_{s,i_{1}} = −c_{s,i_{2}} and hence the equality φ_{s}(z) = 0 reduces to (5). Therefore, P (z) and Q(z) have a common right divisor in the composition algebra by lemma 3.1.

3.2 Reduction

Although condition (19) in general is weaker than condition (2) it turns out that in order to prove that for any collection P (z), a, b, a ≠ b, satisfying some condition R conditions (1) and (2) are equivalent it is often enough to show that for any such a collection condition (1) implies condition (19). Say that a condition R is compositionally stable if for any collection P (z), a, b, a ≠ b, satisfying R such that P (z) = P̃(W (z)) for some P̃(z), W (z) ∈ ℂ[z], deg W (z) > 1, W (a) ≠ W (b), the collection P̃(z), W (a), W (b) also satisfies R. For instance, the following condition is compositional stable: at least one point from a, b is not a critical point of P (z). An other example of a compositional stable condition is the following one: deg P (z) = p^{r}, where p is a prime.

Lemma 3.2 Let R be a compositionally stable condition. Suppose that for any collection P (z), a, b, a ≠ b, satisfying R, condition (1) implies condition (19). Then for any collection P (z), a, b, a ≠ b, satisfying R conditions (1) and (2) are equivalent.

Proof. Let P (z), a, b be a collection satisfying R. Suppose that (1) holds for some q(z) ∈ ℂ[z]. Then it follows from (19) that ℂ(P, Q) is a proper subfield of ℂ(z). Therefore, by the Lüroth theorem

ℂ(P, Q) = ℂ(W_{1})  (20)

for some rational function W_{1}(z), deg W_{1}(z) > 1, and without loss of generality we can assume that W_{1}(z) is a polynomial. To prove the lemma it is enough to show that the equality W_{1}(a) = W_{1}(b) holds.

Observe that equality (20) is equivalent to the statement that

P (z) = P_{1}(W_{1}(z)),  Q(z) = Q_{1}(W_{1}(z))  (21)

for some polynomials P_{1}(z), Q_{1}(z) such that P_{1}(z) and Q_{1}(z) have no a common right divisor in the composition algebra. On the other hand, performing the change of variable z → W_{1}(z) we see that condition (1) is satisfied also for P_{1}(z), Q′_{1}(z), W_{1}(a), W_{1}(b). Therefore, if W_{1}(a) ≠ W_{1}(b), then, taking into account that R is compositionally stable, we find that

P_{1}(z) = P_{2}(W_{2}(z)), Q_{1}(z) = Q_{2}(W_{2}(z))

for some P_{2}(z), Q_{2}(z), W_{2}(z) ∈ ℂ[z] with deg W_{2}(z) > 1 that contradicts the fact that P_{1}(z), Q_{1}(z) have no a common right divisor in the composition algebra.

3.3 Description of some classes of definite polynomials

As a first application of theorem 3.1 and lemma 3.2 we give a simple proof of the following assertion conjectured in [18].

Corollary 3.1 Let P (z), q(z) ∈ ℂ[z], q(z) ≠ 0, a, b ∈ ℂ, a ≠ b. Suppose that P (a) = P (b) = c_{1} and that all the points of the preimage P^{−1}(c_{1}) except possibly a, b are not critical points of P (z). Then conditions (1) and (2) are equivalent.

Proof. Since the chain rule implies that the condition of the corollary is compositionally stable it is enough to show that P (z), Q(z) have a common right divisor in the composition algebra. To establish it observe that Γ_{a,b} can not pass through vertices of λ_{P} of the valency 1 distinct from a, b. Therefore, the condition of the corollary implies that w(1) = 1. It follows now from theorem 3.1 that P (z), Q(z) have a common right divisor in the composition algebra.

A slight modification of the idea used in the proof of corollary (3.1) leads to the following statement.

Corollary 3.2 Let P (z), q(z) ∈ ℂ[z], q(z) ≠ 0, a, b ∈ ℂ, a ≠ b. Suppose that P (a) = P (b) = c_{1} and that for any critical value c of P (z) except possibly c_{1} the preimage P^{−1}(c) contains only one critical point. Then conditions (1) and (2) are equivalent.

Proof. Again, it follows from the chain rule that the condition of the corollary is compositionally stable. Furthermore, observe that the path Γ_{a,b} contains at least one vertex v of a color s ≠ 1. Since Γ_{a,b} can not pass through vertices of the valency 1 distinct from a, b, it follows from the condition of the corollary that the equality w(s) = 1 holds and, therefore, by theorem 3.1, P (z), Q(z) have a common right divisor in the composition algebra.

Finally, we give a new proof of an assertion from the paper [18] which provides some geometric condition for a collection P (z), a, b to be definite. It turns out that this assertion actually also can be regarded as a particular case of theorem 3.1. For a curve M denote by V_{M,∞} the domain from the collection of domains ℂℙ^{1} \ M which contains infinity. For an oriented curve L and points d_{1}, d_{2} ∈ L denote by L_{d_{1},d_{2}} the part of L between d_{1} and d_{2}.

Corollary 3.3 Let P (z), q(z) ∈ ℂ[z], q(z) ≠ 0, a, b ∈ ℂ, a ≠ b. Suppose that P (a) = P (b) = c_{1} and that there exists a curve L connecting points a, b such that c_{1} is a simple point of P (L) and c_{1} ∈ ∂V_{P (L),∞}. Then conditions (1) and (2) are equivalent.

Proof. We will keep the notation introduced in subsection 2.2 and 2.3. Let a^{+} (resp. b^{−}) be a point on L near the point a (resp. b) and let U be a simply connected domain containing no critical values of P (z) such that the sets S \ {c_{1}, c_{2}, ..., ck̃}, P (L_{a,a^{+}}) \ c_{1}, and P (L_{b^{−},b}) \ c_{1} are subsets of U. Recall that there is a natural correspondence between branches P_{i}^{−1}(z), 1 ≤ i ≤ n, of P^{−1}(z) in U and stars of the cactus λ_{P} : branch P_{i}^{−1}(z) maps U on a domain U_{i} containing S_{i}.

Denote by U_{j_{1}} (resp. U_{j_{2}}) the domain containing the point a^{+} (resp. b^{−}). Then by construction the result of the analytic continuation of the element (P_{j_{1}}^{−1}(z), U) along the curve P (L_{a^{+},b^{−}}) is the element (P_{j_{2}}^{−1}(z), U). Let c_{0} be an interior point of U close to c_{1}. Consider a small deformation M of the curve P (L) obtained as follows: change the part of P (L) connecting c_{1} and P (a^{+}) to an arc γ^{+} ⊂ U connecting c_{0} with P (a^{+}) and, similarly, change the part of P (L) connecting P (b^{−}) and c_{1} to an arc γ^{−} ⊂ U connecting P (b^{−}) and c_{0} (see Fig. 2).

Let now l_{M} = l_{i_{1}}^{j_{1}}l_{i_{2}}^{j_{2}} ... l_{i_{r}}^{j_{r}} be the image of the curve M in the fundamental group π_{1}(X, c_{0}), where X = ℂℙ^{1} \ {c_{1}, c_{2}, c_{3}, ... c_{k}}. Since the result of the analytic continuation of the element (P_{j_{1}}^{−1}(z), U) along the curve M is still the element (P_{j_{2}}^{−1}(z), U), the final element of the chain of stars

Ω =< S_{j_{1}} , S_{g_{i_{1}}^{j_{1}} (j)} , S_{g_{i_{1}}^{j_{1}} g_{i_{2}}^{j_{2}} (j)} , S_{g_{i_{1}}^{j_{1}} g_{i_{2}}^{j_{2}} g_{i_{3}}^{j_{3}} (j)} , ... , S_{g_{i_{1}}^{j_{1}} g_{i_{2}}^{j_{2}} ... g_{ir}^{jr} (j)} >

is the star S_{j_{2}}. In particular, the path Γ_{a,b} is contained in Ω. Since c_{1} ∈ V_{M,∞}, the loop l_{1} does not appear among the loops l_{i_{1}}, l_{i_{2}}, ... l_{i_{r}}. Therefore, the common vertex of any two successive stars in the chain Ω is not contained in the set P^{−1}(c_{1}). In particular, among of vertices of Γ_{a,b} there are no preimages of c_{1} distinct from a, b and hence w(1) = 1 on Γ_{a,b}.

To finish the proof notice that the condition of the corollary is compositionally stable. Indeed, if L is a curve connecting points a, b such that c_{1} = P (a) = P (b) is a simple point of P (L) and c_{1} ∈ ∂V_{P (L),∞}, then W (L) is a curve connecting points W (a), W (b) such that c_{1} = P̃(W (a)) = P̃(W (b)) is a simple point of P̃(W (L)) and c_{1} ∈ ∂V_{P̃(W (L)),∞}.

4 Functional equations (6) and (7)

4.1 Derivation of (6) and (7)

Let U be a simply connected domain containing no critical values of P (z) such that S\{c_{1}, c_{2}, ..., ck̃} ⊂ U. Denote by P_{a_{1}}^{−1}(z), P_{a_{2}}^{−1}(z), ..., P_{a_{da}}^{−1} (z) (resp. P_{b_{1}}^{−1}(z), P_{b_{2}}^{−1}(z), ... , P_{b_{d_{b}}}^{−1}(z)) the branches of P^{−1}(z) in U which map points close to P (a) (resp. P (b)) to points close to a (resp. b). In particular, d_{a} (resp. d_{b}) equals the multiplicity of the point a (resp. b) with respect to P (z). It was shown in [15] for P (a) = P (b) and in [18] in general case that condition (1) implies equality (6) or system (7), where as above Q(z) = ∫ q(z)dz is normalized by the condition Q(a) = Q(b) = 0.

For the sake of self-containedness of this paper below we provide a short derivation of (6), (7) from theorem 2.1.

Proposition 4.1 Suppose that condition (1) holds. Then, if P (a) = P (b), equation (6) holds in U. Furthermore, if P (a) ≠ P (b), then system (7) holds in U.

Proof. Suppose first that P (a) = P (b) = c_{1}. Examine the relation

φ_{1}(z) = ∑_{i=1}^{n} f_{1,i}Q(P_{i}^{−1}(z)) = 0.

Let i, 1 ≤ i ≤ n, be an index such that f_{1,i} ≠ 0 and let x be a vertex of the star S_{i} such that P (x) = c_{1}. Observe that if x ≠ a, x ≠ b, then there exists an index ĩ such that x is also a vertex of the star Sĩ and f_{1,ĩ} = −f_{1,i}. Furthermore, we have ĩ = g_{1}^{j}(i) for some natural number j. Therefore, φ_{1}(z) has the form

φ_{1}(z) = −Q(P_{i_{a}}^{−1}(z))+
Q(P_{i_{1}}^{−1}(z)) − Q(P_{g_{1}^{j_{1}} (i_{1})}^{−1}(z)) + ... + Q(P_{i_{r}}^{−1}(z)) − Q(P_{g_{1}^{jr} (i_{r})}^{−1}(z))
+Q(P_{i_{b}}^{−1}(z)) = 0,

where i_{a} (resp. i_{b}) is the index such that a ⊂ S_{i_{a}} (resp. b ⊂ S_{i_{b}}), i_{1}, i_{2}, ... i_{r} are some other indices and j_{1}, j_{2}, ... j_{r} are some natural numbers.

Let n_{1} be the order of the element g_{1} in the group G_{P} . For each s, 0 ≤ s ≤ n_{1} − 1, the equality

−Q(P_{g_{1}^{s}(i_{a})}^{−1}(z))+
Q(P_{g_{1}^{s}(i_{1})}^{−1}(z)) − Q(P_{g_{1}^{j_{1}+s}(i_{1})}^{−1}(z)) + ... + Q(P_{g_{1}^{s}(i_{r})}^{−1}(z)) − Q(P_{g_{1}^{jr +s}(i_{r})}^{−1}(z))
+Q(P_{g_{1}^{s}(i_{b})}^{−1}(z)) = 0

holds by the analytic continuation of the equality φ_{1}(z) = 0. Summing these equalities and taking into account that for any i, 1 ≤ i ≤ n, and any natural number j we have:

∑_{s=0}^{n_{1}−1} Q(P_{g_{1}^{s}(i)}^{−1}(z)) = ∑_{s=0}^{n_{1}−1} Q(P_{g_{1}^{j+s}(i)}^{−1}(z)),

we obtain equality (6).

In the case when P (a) ≠ P (b) the proof is similar: if P (a) = c_{1}, P (b) = c_{2}, then one must examine relations φ_{1}(z) = 0 and φ_{2}(z) = 0.

Note that if points a, b are not critical points of P (z), then (6) reduces to (5) while (7) leads to the equality q(z) ≡ 0. In view of lemmas 3.1 and 3.2 this implies immediately the following result from [9] (see also [15],[18]).

Corollary 4.1 Let P (z), q(z) ∈ ℂ[z], q(z) ≠ 0, a, b ∈ ℂ, a ≠ b. Suppose that a, b are not critical points of P (z). Then conditions (1) and (2) are equivalent.

4.2 Relations between branches of Q(P^{−1}(z))

In this subsection we examine how linear relations between branches of Q(P^{−1}(z)) over ℂ reflect on the structure of coefficients of the Puiseux expansion of Q(P^{−1}(z)) near infinity.

Let P (z) be a non-constant polynomial of degree n and let z_{0} ∈ ℂ be a non-critical value of P (z). If |z_{0}| is sufficiently large then in a neighborhood U_{z_{0}} of z_{0} each branch of P^{−1}(z) can be represented by a Puiseux series centered at infinity. More precisely, if P_{0}^{−1}(z) is a fixed branch of P^{−1}(z) near z_{0} then in U_{z_{0}} we have:

P_{0}^{−1}(z) = ∑_{k=−1}^{∞} v_{k}z^{− \frac{k}{n}} ,  v_{k} ∈ ℂ,  ε_{n} = exp(2πi/n),

where z^{\frac{1}{n}} is a branch of the algebraic function which is inverse to z^{n} in U_{z_{0}}. If l is a loop around infinity then the result of the analytic continuation of the branch P_{0}^{−1}(z) along l^{j}, 0 ≤ j ≤ n − 1, is represented by the series

P_{j}^{−1}(z) = ∑_{k=−1}^{∞} v_{k}ε_{n}^{jk}z^{− \frac{k}{n}} .  (22)

The numeration of branches of P^{−1}(z) near z_{0} defined by equation (22) is called canonical. Clearly, such a numeration depends on the choice of P_{0}^{−1}(z). Nevertheless, any canonical numeration induces the same cyclic ordering of branches of P^{−1}(z) in U_{z_{0}}. This cyclic ordering also will be called canonical. For any nonzero polynomial Q(z), deg Q(z) = m, the composition Q(P_{j}^{−1}(z)), 0 ≤ j ≤ n−1, is represented near z_{0} by the series (8) obtained by the substitution of series (22) in Q(z).

Let U be a simply-connected domain containing no critical values of P (z) such that some linear combination of branches of Q(P^{−1}(z)) over ℂ identically vanishes in U. Considering in case of necessity a bigger domain we can suppose without loss of generality that ∞ ∈ ∂U. Then series (22) converge in a domain V ⊂ U. Furthermore, we can assume that the numeration of branches of P^{−1}(z) in U is induced by a canonical numeration of branches of P^{−1}(z) in V. If equality

∑_{i=0}^{n−1} f_{j}Q(P_{j}^{−1}(z)) = 0,  f_{j} ∈ ℂ,  (23)

holds in U, then substituting in (23) expansions (8) we see that (23) reduces to the system

∑_{j=0}^{n−1} f_{j}u_{k}ε_{n}^{kj} = 0, k ≥ −m.

Introducing the notation F (z) = ∑_{j=0}^{n−1} f_{j}z^{j} and summing up we get:

Lemma 4.1 The equality (23) holds in U if and only if for any k ≥ −m either u_{k} = 0 or F (ε_{n}^{k}) = 0.

In particular, since all u_{k} can not vanish and deg F (z) < n, the following statement is true.

Corollary 4.2 If equality (23) holds in U then F (ε_{n}^{r}) = 0 for at least one r, 0 ≤ r ≤ n − 1. On the other hand, for at least one r, 0 ≤ r ≤ n − 1, the equality u_{k} = 0 holds whenever k ≡ r mod n.

4.3 Lemma about monodromy groups of polynomials

In order to relate (6), (7) with coefficients of the Puiseux expansion of Q(P^{−1}(z)) near infinity we are going to examine which roots of unity can be roots of the corresponding polynomial

r(z) = \frac{1}{d_{a}} ∑_{s=1}^{d_{a}} z^{a_{s}} − \frac{1}{d_{b}} ∑_{s=1}^{d_{b}} z^{b_{s}},  (24)

or common roots of the corresponding pair of polynomials

r_{1}(z) = \frac{1}{d_{a}} ∑_{s=1}^{d_{a}} z^{a_{s}},  r_{2}(z) = \frac{1}{d_{b}} ∑_{s=1}^{d_{b}} z^{b_{s}}.  (25)

For this propose we establish now a geometric property of monodromy groups of polynomials which concerns the mutual arrangement of indices a_{1}, a_{2}, ..., a_{d_{a}} and b_{1}, b_{2}, ... , b_{d_{b}} under assumption that the numeration of branches is canonical.

Let P (z) ∈ ℂ[z], deg P (z) = n, a, b ∈ ℂ, a ≠ b. Let U be a simply-connected domain containing no critical values of P (z) such that P (a), P (b), ∞ ∈ ∂U. Fix a canonical numeration of branches of P^{−1}(z) in U and let P_{u_{1}}^{−1}(z), P_{u_{2}}^{−1}(z), ..., P_{u_{da}}^{−1} (z) (resp. P_{v_{1}}^{−1}(z), P_{v_{2}}^{−1}(z), ..., P_{v_{d_{b}}}^{−1}(z)) be the branches of P^{−1}(z) in U which map points close to P (a) (resp. P (b)) to points close to the point a (resp. b) numbered by means of this numeration. The lemma below describes the mutual position on the unit circle of the sets V (a) = {ε_{n}^{a_{1}}, ε_{n}^{a_{2}}, ..., ε_{n}^{a_{da}} } and V (b) = {ε_{n}^{b_{1}}, ε_{n}^{b_{2}}, ..., ε_{n}^{b_{d_{b}}} }, where ε_{n} = exp(2πi/n).

Let us introduce the following definitions. Say that two sets of points X, Y on the unit circle S_{1} are disjointed if there exist s_{1}, s_{2} ∈ S_{1} such that all points from X are on the one of two connected components of S_{1} \ {s_{1}, s_{2}} while all points from Y are on the other one. Say that X, Y are almost disjointed if X ∩Y consists of a single point s_{1} and there exists a point s_{2} ∈ S_{1} such that all points from X \ s_{1} are on the one of two connected components of S_{1} \ {s_{1}, s_{2}} while all points from Y \ s_{1} are on the other one.

Monodromy Lemma. The sets V (a) and V (b) are almost disjointed. Furthermore, if P (a) = P (b) then V (a) and V (b) are disjointed.

Proof. Consider first the case when P (a) = P (b). Let M ⊂ U be a simple curve connecting points P (a) = P (b) and ∞. Consider the preimage P^{−1}{M} of M under the map P (z) : ℂℙ^{1} → ℂℙ^{1}. It is convenient to consider P^{−1}{M} as a bicolored graph Ω embedded into the Riemann sphere: the black vertices of Ω are preimages of P (a) = P (b), the unique white vertex is the preimage of ∞, and the edges of Ω are preimages of M (see Fig. 3). Since the multiplicity of the vertex ∞ equals n and Ω has n edges, Ω is connected. The edges of Ω are identified with branches of P^{−1}(z) in U as follows: to a branch P_{k}^{−1}(z), 1 ≤ k ≤ n, corresponds the edge e_{k} such that P_{k}^{−1}(z) maps M \ {P (a), ∞} into e_{k}. In particular, the canonical cyclic ordering of branches of P^{−1}(z) in U induces a cyclic ordering on edges of Ω.

For any vertex v of Ω the orientation of ℂℙ^{1} induces a natural cyclic ordering on edges of Ω adjacent to v. In particular, taking v = ∞, we obtain a cyclic ordering on edges of Ω. Clearly, this cyclic ordering coincides with that induced by the canonical cyclic ordering of branches of P^{−1}(z) in U. Let E_{a} = {e_{a_{1}}, e_{a_{2}}, ... , e_{a_{da}} } (resp. E_{b} = {e_{b_{1}}, e_{b_{2}}, ... , e_{b_{d_{b}}} }) be the union of edges of Ω which are adjacent to the vertex a (resp. b). Let D be the domain from the collection of domains ℂℙ^{1} \ E_{a} which contains point b and let e_{s}, e_{t} ∈ E_{a} be the edges which bound D. Clearly, all the edges from E_{a} are contained in ℂℙ^{1} \ D. Therefore, the lemma is equivalent to the following statement: the domain D contains e_{h} \ ∞ for all e_{h} ∈ E_{b}. But the last statement is a corollary of the Jordan theorem since an edge e_{h} ∈ E_{b} can intersect e_{s} or e_{t} only at infinity.

In the case when P (a) ≠ P (b) the proof is modified as follows. Divide the boundary of U into three parts M_{1}, M_{2}, M_{3}, where M_{1} connects the point ∞ with the point P (a), M_{2} connects the point ∞ with the point P (b), and M_{3} connects the point P (a) with the point P (b). Consider now P^{−1}{∂U} as a graph Ω embedded into the Riemann sphere. The vertices of Ω are divided into three groups: the first one consists of vertices that are preimages of ∞, the second one consists of vertices that are preimages of P (b), and the third one consists of vertices that are preimages of P (a). Similarly, the edges of Ω also are divided into three groups: the first one consists of edges that are preimages of M_{1}, the second one consists of edges that are preimages of M_{2}, and the third one consists of edges that are preimages of M_{3}. Finally, the faces of Ω are divided into two groups: the first one consists of faces that are preimages of U and the second one consists of faces that are preimages of ℂℙ^{1} \ \overline{U} (see Fig. 4).

The faces from the first group are identified with branches of P^{−1}(z) in U as follows: to a branch P_{k}^{−1}(z), 1 ≤ k ≤ n, corresponds the face f_{k} such that P_{k}^{−1}(z) maps bijectively U on f_{k} \∂f_{k}. The edges from the corresponding groups which bound f_{k} will be denoted by e_{k}^{1}, e_{k}^{2}, e_{k}^{3} correspondingly. Note that in a counterclockwise direction around infinity the edge e_{k}^{1}, 1 ≤ k ≤ n, is followed by the edge e_{k}^{2}. The canonical cyclic ordering of branches of P^{−1}(z) in U induces a cyclic ordering of faces of Ω belonging to the first group of faces. Clearly, this ordering coincides with the natural ordering induced by the orientation of ℂℙ^{1}.

Let E_{a}^{1} = {e_{a_{1}}^{1}, e_{a_{2}}^{1}, ... , e_{a_{da}}^{1} } (resp. E_{b}^{2} = {e_{b_{1}}^{2}, e_{b_{2}}^{2}, ... , e_{b_{d_{b}}}^{2} }) be the union of edges from the first (resp. the second) group Ω which are adjacent to the vertex a (resp. b). Let D be the domain from the collection of domains ℂℙ^{1} \E_{a}^{1} which contains point b. Once again the Jordan theorem implies that all the edges from E_{a}^{1} are contained in ℂℙ^{1} \ D while D contains e_{h}^{2} \ ∞ for all e_{h}^{2} ∈ E_{b}^{2}. Taking into account that for any k, 1 ≤ k ≤ n, the edge e_{k}^{1} is followed by e_{k}^{2} this fact implies that V (a) and V (b) are almost disjointed. Note that, in contrast to the case when P (a) = P (b), now the sets V (a) and V (b) can have a non-empty intersection consisting of a single element.

4.4 On coefficients of Puiseux expansion of Q(P^{−1}(z))

In this subsection we deduce from the monodromy lemma the following important property of the Puiseux expansion (8) for pairs P (z), Q(z) satisfying (6), (7).

Theorem 4.1 Let P (z), Q(z) ∈ ℂ[z], deg P (z) = n, a, b ∈ ℂ, a ≠ b. Suppose that (6) or (7) holds. Then u_{k} = 0 for any k such that GCD(k, n) = 1.

Proof. Suppose first that P (a) = P (b). Then lemma 4.1 implies that u_{k} = 0 whenever the number ε_{n}^{k} is not a root of the polynomial (24). Let us show that if GCD(k, n) = 1 then the equality r(ε_{n}^{k}) = 0 is impossible. Indeed, if (k, n) = 1, then ε_{n}^{k} is a primitive n-th root of unity. Since the n-th cyclotomic polynomial Φ_{n}(z) is irreducible over ℤ, the equality r(ε_{n}^{k}) = 0 implies that Φ_{n}(z) divides r(z) in the ring ℤ[z]. Therefore, the primitive n-th root of unity ε_{n} = exp(2πi/n) also is a root of r(z) and hence the equality

∑_{s=1}^{d_{a}} ε_{n}^{a_{s}}/d_{a} = ∑_{s=1}^{d_{b}} ε_{n}^{b_{s}}/d_{b}

holds. The last equality is equivalent to the statement that the mass centers of the sets V (a) and V (b) coincide. But this contradicts to the monodromy lemma. Indeed, the mass center of a system of points in ℂ is inside of the convex envelope of this system and therefore the mass centers of disjointed sets must be distinct.

If P (a) ≠ P (b) then, similarly, the inequality u_{k} ≠ 0 for GCD(k, n) = 1 implies that

∑_{s=1}^{d_{a}} ε_{n}^{a_{s}}/d_{a} = 0, ∑_{s=1}^{d_{b}} ε_{n}^{b_{s}}/d_{b} = 0.

But this again contradicts the monodromy lemma. Indeed, the fact that the sets V (a) and V (b) are almost disjointed implies that at least one from these sets is contained in an open half plane bounded by a line passing through the origin and therefore the mass center of this set is distinct from zero.

Corollary 4.3 Let P (z), q(z) ∈ ℂ[z], q(z) ≠ 0, deg P (z) = n, deg Q(z) = m, a, b ∈ ℂ, a ≠ b. Suppose that (1) holds. Then GCD(m, n) > 1.

Proof. Since in expansions (22) the coefficient v_{−1} is distinct from zero, the coefficient u_{−m} = v_{−1}^{m} in expansions (8) is distinct from zero. Since (1) implies (6) or (7) by proposition 4.1 it follows from theorem 4.1 that GCD(m, n) > 1.

Notice that theorem 4.1 agrees with conjecture (3). Indeed, if

Q(z) = Q̃_{1}(W_{1}(z)) + Q̃_{1}(W_{1}(z)) + ... + Q̃_{r}(W_{r}(z)),  (26)

where W_{1}(z), W_{2}(z), ..., W_{r}(z) are (non-trivial) right divisors of P (z) in the composition algebra,

P (z) = P̃_{1}(W_{1}(z)) = P̃_{2}(W_{2}(z)) = ... = P̃_{r}(W_{r}(z)),

then the expansion (8) has the form

Q(P^{−1}(z)) = Q̃_{1}(P̃_{1}^{−1}(z)) + Q̃_{2}(P̃_{2}^{−1}(z)) + ... + Q̃_{r}(P̃_{r}^{−1}(z)).

Since deg P̃_{j}(z) < n, 1 ≤ j ≤ r, it follows easily that u_{k} = 0 for any k such that GCD(k, n) = 1. Conjecturally, vice versa, equalities u_{k} = 0 for all k with GCD(k, n) = 1 imply that Q(z) has form (26) at least under some additional assumptions. We plan to discuss this topic in another paper.

5 Further description of definite polynomials

5.1 Case when a or b is not a critical point of P (z)

As a first application of the Puiseux expansions technique we provide in this subsection the following generalization of corollary 4.1.

Theorem 5.1 Let P (z), q(z) ∈ ℂ[z], q(z) ≠ 0, a, b ∈ ℂ, a ≠ b. Suppose that at least one from points a and b is not a critical point of the polynomial P (z). Then conditions (1) and (2) are equivalent.

Proof. Since the condition of the theorem is compositionally stable it follows from lemmas 3.2, 3.1 that we only must show that equality (5) holds. To be definite suppose that the point a is not a critical point of P (z). By proposition 4.1 either the system

Q(p_{a_{1}}^{−1}(z)) = 0,  ∑_{s=1}^{d_{b}} Q(p_{b_{s}}^{−1}(z)) = 0  (27)

or the equality

Q(p_{a_{1}}^{−1}(z)) = \frac{1}{d_{b}} ∑_{s=1}^{d_{b}} Q(p_{b_{s}}^{−1}(z))  (28)

holds. Nevertheless, since the first equation of system (27) leads to the equality q(z) ≡ 0, we only must consider equation (28).

Applying lemma 4.1 we see that for any k such that u_{k} ≠ 0 the equality

d_{b}(ε_{n}^{k})^{a_{1}} = ∑_{s=1}^{d_{b}}(ε_{n}^{k})^{b_{s}}

holds. The triangle inequality implies that this is possible only if

(ε_{n}^{k})^{a_{1}} = (ε_{n}^{k})^{b_{1}} = (ε_{n}^{k})^{b_{2}} = ... = (ε_{n}^{k})^{b_{d_{b}}} .

Therefore,

Q(p_{a_{1}}^{−1}(z)) = Q(p_{b_{1}}^{−1}(z)) = Q(p_{b_{2}}^{−1}(z)) = ... = Q(p_{b_{ds}}^{−1} (z)).

5.2 Case when deg P (z) = p^{n}

In this subsection we deduce from theorem 4.1 the solution of the polynomial moment problem in the case when deg P (z) = p^{r} for p prime.

Theorem 5.2 Let P (z), q(z) ∈ ℂ[z], q(z) ≠ 0, a, b ∈ ℂ, a ≠ b. Suppose that deg P (z) = p^{r}, where p is a prime number. Then conditions (1) and (2) are equivalent.

Proof. Again, since the condition of the theorem is compositionally stable, it is enough to show that (5) holds. Consider expansion (8). By theorem 4.1 the equality u_{k} = 0 holds for any k with GCD(k, p^{r}) = 1. Show that this fact implies the equality

Q(P_{j}^{−1}(z)) = Q(P_{j+p^{r−1}}^{−1}(z))

for any j, 0 ≤ j ≤ n − 1. Indeed, we have:

Q(P_{j}^{−1}(z)) − Q(P_{j+p^{r−1}}^{−1}(z)) = ∑_{k=−m}^{∞} w_{k}z^{− \frac{k}{n}} ,

where

w_{k} = u_{k}(ε_{p^{r}}^{jk} − ε_{p^{r}}^{(j+p^{r−1})k}).

If GCD(k, p^{r}) = 1 then u_{k} = 0 and hence w_{k} = 0. Otherwise, k = pk̃ for some k̃ ∈ ℤ. Therefore,

ε_{p^{r}}^{(j+p^{r−1})k} = ε_{p^{r}}^{jk}ε_{p^{r}}^{p^{r} k̃} = ε_{p^{r}}^{jk}

and hence again w_{k} = 0.

5.3 Case when P (z) is indecomposable

Theorems 2.1, 5.2 allow us to give a short proof of the theorem proved in [15],

[17] which describes solutions to (1) in case when P (z) is indecomposable that is can not be represented as a composition P (z) = P_{1}(P_{2}(z)) with non-linear polynomials P_{1}(z), P_{2}(z).

Theorem 5.3 Let P (z), q(z) ∈ ℂ[z], q(z) ≠ 0, a, b ∈ ℂ, a ≠ b. Suppose that P (z) is indecomposable. Then conditions (1) and (2) are equivalent. In more details, Q(z) is a polynomial in P (z) and P (a) = P (b).

Proof. Once again we only must prove that (5) holds. Suppose the contrary that is that all Q(P_{i}^{−1}(z)), 1 ≤ i ≤ n, where n = deg P (z) are different; then the monodromy group G of the algebraic function Q(P^{−1}(z)) obtained by the complete analytic continuation of Q(P_{i}^{−1}(z)), 1 ≤ i ≤ n, coincides with that of P^{−1}(z). Since P (z) is indecomposable, G is primitive by the Ritt theorem [19]. Since for the case when n = deg P (z) is a prime number the statement follows from theorem 5.2 we can suppose that n is a composite number. By the Schur theorem (see e.g. [22], Th. 25.3) a primitive permutation group of composite degree n which contains an n-cycle is doubly transitive.

Recall now the following fact: roots α_{i}, 1 ≤ i ≤ n, of an irreducible algebraic equation over a field k of characteristic zero with doubly transitive Galois group can not satisfy any relation

∑_{i=1}^{n} c_{i}α_{i} = 0, c_{i} ∈ k,

except the case when c_{1} = c_{2} = ... = c_{n} (see [11], Proposition 4, or, in the context of algebraic functions, [15], lemma 2). Since the monodromy group of an algebraic function coincides with the Galois group of the equation over ℂ(z) which defines this function, it follows that if all Q(P_{i}^{−1}(z)), 1 ≤ i ≤ n, are different, then equality (23) is possible only when

f_{1} = f_{2} = ... = f_{n}.  (29)

On the other hand, for any non-trivial equation φ_{s}(z) = 0 appeared in theorem 2.1 the equality (29) is impossible by construction. This contradiction completes the proof.

6 Solution of the polynomial moment problem for polynomials of degree less than 10

In this section we provide a complete solution of the polynomial moment problem for polynomials of degree less than 10.

For an extended cactus λ̃_{P} and a path Γ_{a,b} define the skeleton Γ̂_{a,b} of Γ_{a,b} as follows. Draw the path Γ_{a,b} separately from the graph λ̃_{P} and erase all its white vertices. Number the edges of the obtained graph Γ̂_{a,b} so that the number of an edge e_{k} coincides with the number of the star S_{k} of λ̃_{P} for which e_{k} ⊂ S_{k}. The number of edges of Γ̂_{a,b} is called the length l(Γ̂_{a,b}) of Γ̂_{a,b}. For example, the skeleton Γ̂_{a,b} of the path Γ_{a,b} from Fig. 1 is shown on Fig. 5; here l(Γ̂_{a,b}) = 4.

Figure 5

Theorem 6.1 Let P (z), q(z) ∈ ℂ[z], q(z) ≠ 0, a, b ∈ ℂ, a ≠ b, satisfy (1). Suppose that deg P (z) < 10. Then either condition (2) holds or there exist linear functions L_{1}(z), L_{2}(z) such that

L_{2}(P (L_{1}(z))) = T_{6}(z), L_{1}^{−1}(a) = −\sqrt{3}/2, L_{1}^{−1}(b) = \sqrt{3}/2,

and

Q(L_{1}(z)) = A(T_{3}(z)) + B(T_{2}(z))

for some A(z), B(z) ∈ ℂ[z].

Proof. First of all observe that any natural number n < 10 distinct from 6 is either a prime number or a degree of a prime number. Therefore, it follows from theorem 5.2 that it suffices to consider the case when deg P (z) = 6. Furthermore, in view of theorem 5.1 we can suppose that the points a, b are critical points of P (z). Finally notice that in order to prove that condition (2) holds for P (z), q(z) satisfying (1) with deg P (z) = 6 it is enough to establish equality (19). Indeed, if W (a) ≠ W (b) in (19) then performing the change of variable z → W (z) we see that (1) holds for P̃(z), Q̃(z), W (a), W (b). If deg W (z) equals 3 or 2, then it follows from theorem 5.3 that Q̃(z) = R(P̃(z)) for some R(z) ∈ ℂ[z] and P̃(W (a)) = P̃(W (b)). Therefore, (2) holds with W (z) = P (z), Q̃(z) = R(z). On the other hand, if deg W (z) = 6 in (19) then necessary W (a) = W (b) since otherwise Q̃(z) would be orthogonal to all powers of z on the segment W (a), W (b). In particular, in view of lemma 3.1, we see that in order to prove that conditions (1) and (2) are equivalent it is enough to establish (5).

Since deg P (z) = 6, clearly l(Γ̂_{a,b}) ≤ 6. Moreover, since the points a, b are critical points of P (z), the valency of the corresponding vertices of λ̃_{P} is at least 2, and, therefore, actually l(Γ̂_{a,b}) ≤ 4. Consider all possible cases. First of all observe that the equality l(Γ̂_{a,b}) = 1 is impossible. Indeed, in this case theorem 2.1 implies that Q(P_{i}^{−1}(z)) = 0, where i is the number of the unique edge of Γ̂_{a,b}, and therefore q(z) ≡ 0. Furthermore, if l(Γ̂_{a,b}) = 2 then, since adjacent vertices of Γ̂_{a,b} have different colors, Γ̂_{a,b} can be of one from the following two forms shown on Fig. 6.

Figure 6

In both cases for the middle vertex y we have w(y) = 1. Therefore by theorem 3.1 equality (19) holds and hence conditions (1) and (2) are equivalent. Observe, however, that the first configuration shown on Fig. 6 is actually not realizable since (2) implies that P (a) = P (b).

Consider now the case when l(Γ̂_{a,b}) = 3. It is not difficult to see that in this case either again w(y) = 1 for some color y or Γ̂_{a,b} has the form shown on Fig. 7.

Figure 7

Let us examine the last case. Since for the skeleton shown on Fig. 7 we have P (a) ≠ P (b), it follows from proposition 4.1 that system 7 holds. Furthermore, the equality deg P (z) = 6 implies that for at least one vertex s ∈ {a, b} the following two conditions are satisfied: the multiplicity of s equals 2 and the connectivity component of λ̃_{P} \ s which does not contain Γ_{a,b} consists of a unique star. To be definite suppose that s = a. Then, in notation of 2.2, the first condition implies that

∑_{s=1}^{d_{a}} Q(P_{a_{s}}^{−1}(z)) = Q(P_{i_{1}}^{−1}(z)) + Q(P_{g_{x}(i_{1})}^{−1}(z)) = 0  (30)

and the second one that g_{y}(g_{x}(i_{1})) = g_{x}(i_{1}). Therefore, the analytic continuation of (30) along the loop l_{y} leads to the equality

Q(P_{i_{2}}^{−1}(z)) + Q(P_{g_{x}(i_{1})}^{−1}(z)) = 0.  (31)

Now equalities (30),(31) imply that Q(P_{i_{1}}^{−1}(z)) = Q(P_{i_{2}}^{−1}(z)) and we conclude as above that the configuration shown on Fig. 7 is not realizable.

Consider finally the case when l(Γ̂_{a,b}) = 4. Since Γ̂_{a,b} has 5 vertices, either w(y) = 1 for some color y or Γ̂_{a,b} is two-colored. In the last case Γ̂_{a,b} has the form shown on Fig. 8

Figure 8

and the corresponding cactus λ̃_{P} is a 6-chain (the cactus with 6 stars of the maximal diameter). Furthermore, since deg P (z) = 6, it follows from the Riemann-Hurwitz formula that

∑_{z∈ℂℙ^{1}}(mult_{z}P − 1) = 10.

Since mult_{∞}P − 1 = 5 and the combinatorics of λ̃_{P} imply that

∑_{P (z)=c_{x}}(mult_{z}P − 1) = 3,  ∑_{P (z)=c_{y}}(mult_{z}P − 1) = 2,

we conclude that P (z) has only two finite critical values c_{x}, c_{y}.

It follows from the Riemann existence theorem (see e.g. [12]) that a complex polynomial with given critical values is defined by its cactus up to a linear change of variable. On the other hand, it is easy to see using the formula T_{n}(cos φ) = cos nφ that T_{n}(z) has only two critical values −1, 1 and that all critical points of T_{n}(z) are simple, Therefore, the corresponding cactus is a chain. In particular, for P (z) = T_{6}(z) the corresponding cactus realized as the preimage of the segment [−1, 1] (considered as a star connecting 0 with points 1 and −1) has the form shown on Fig. 9 (white vertices are omitted).

Figure 9

Therefore, if we choose linear functions L_{1}(z), L_{2}(z) such that:

L_{1}^{−1}(a) = −\sqrt{3}/2, L_{1}^{−1}(b) = \sqrt{3}/2, L_{2}(c_{x}) = −1, L_{2}(c_{y}) = 1,

the polynomial L_{2}(P (L_{1}(z))) will be equal T_{6}(z).

Finally, the last assertion of the theorem follows from the main result of the paper [16] where all solutions to (1) for P (z) = T_{n}(z) were described.

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