Finite Blaschke products and the construction of rational $\Gamma$-inner functions

Let \[ \Gamma = \{(z+w, zw): |z|\leq 1, |w|\leq 1\} \subset \mathbb{C}^2. \] A $\Gamma$-inner function is defined to be a holomorphic map $h$ from the unit disc $\mathbb{D}$ to $\Gamma$ whose boundary values at almost all points of the unit circle $\mathbb{T}$ belong to the distinguished boundary $b\Gamma$ of $\Gamma$. A rational $\Gamma$-inner function $h$ induces a continuous map $h|_\mathbb{T}$ from the unit circle to $b\Gamma$. The latter set is topologically a M\"obius band and so has fundamental group $\mathbb{Z}$. The {\em degree} of $h$ is defined to be the topological degree of $h|_\mathbb{T}$. In a previous paper the authors showed that if $h=(s,p)$ is a rational $\Gamma$-inner function of degree $n$ then $s^2-4p$ has exactly $n$ zeros in the closed unit disc $\mathbb{D}^-$, counted with an appropriate notion of multiplicity. In this paper, with the aid of a solution of an interpolation problem for finite Blaschke products, we explicitly construct the rational $\Gamma$-inner functions of degree $n$ with the $n$ zeros of $s^2-4p$ and the corresponding values of $s$, prescribed.

A Γ-inner function is a holomorphic map h from the unit disc D to Γ whose boundary values at almost all points of the unit circle T (with respect to Lebesgue measure) belong to bΓ. The Γ-inner functions constitute a natural analog of the inner functions of A. Beurling [12], which play a central role in the function theory of the unit disc. For example, it was known to Nevanlinna and Pick that an npoint interpolation problem for functions in the Schur class is solvable if and only if it is solvable by a rational inner function of degree at most n. Likewise, every n-point interpolation problem for functions in the class Hol(D, Γ) of holomorphic maps from D to Γ, if solvable, has a rational Γ-inner solution (for example, [22,Theorem 4.2]). Here, the degree of a rational Γ-inner function h is defined to be the topological degree of the restriction of h mapping T continuously to bΓ. Since bΓ is homeomorphic to a Möbius band, its fundamental group is Z, and so the degree of h is an integer; it will be denoted by deg(h).
We shall address the analog for rational Γ-inner functions of a problem about rational inner functions solved by W. Blaschke [17]. The Argument Principle tells us that a rational inner function ϕ of degree n has exactly n zeros in D, counted with multiplicity, from which fact one deduces that ϕ is a finite Blaschke product where |c| = 1 and α 1 , . . . , α n are the zeros of ϕ. In similar fashion, we should like to write down, as explicitly as possible, the general rational Γ-inner function of degree n. It was shown in [3] that if h = (s, p) is a rational Γ-inner function of degree n then s 2 − 4p has exactly n zeros in the closed unit disc D − , counted with an appropriate notion of multiplicity. The n zeros of s 2 − 4p can be regarded as analogs of the α j for present purposes.
The variety plays a special role in the function theory of Γ: it is called the royal variety. For a rational Γ-inner function h = (s, p), the zeros of s 2 − 4p in D − are the points λ such that h(λ) ∈ R; we shall call them the royal nodes of h. If σ ∈ D − is a royal node of h then h(σ) = (−2η, η 2 ) for some η ∈ D − ; we call η the royal value of h corresponding to the royal node σ. Let us formalise the problem of describing the general rational Γ-inner function in terms of its royal nodes and values. Problem 1.1. Given distinct points σ 1 , . . . , σ n in D − and values η 1 , . . . , η n in D − find if possible a rational Γ-inner function h of degree n such that h(σ j ) = (−2η j , η 2 j ) for j = 1, . . . , n. The results of this paper show that there is a close connection between Problem 1.1 and an n-point interpolation problem for finite Blaschke products of degree n in which there are interpolation nodes in both D and T and in which tangential information is specified at interpolation nodes in T. To formulate this problem we introduce some terminology. Definition 1.2. Let n ≥ 1 and 0 ≤ k ≤ n. By Blaschke interpolation data we mean a triple (σ, η, ρ) where (1) σ = (σ 1 , σ 2 , . . . , σ n ) is an n-tuple of distinct points such that σ j ∈ T for j = 1, . . . , k and σ j ∈ D for j = k + 1, . . . , n; (2) η = (η 1 , η 2 , . . . , η n ) where η j ∈ T for j = 1, . . . , k and η j ∈ D for j = k + 1, . . . , n; (3) ρ = (ρ 1 , ρ 2 , . . . , ρ k ) where ρ j > 0 for j = 1, . . . , k.
For such data the Blaschke interpolation problem with data (σ, η, ρ) is the following: Problem 1.3. Find if possible a rational inner function ϕ on D (that is, a finite Blaschke product) of degree n with the properties ϕ(σ j ) = η j for j = 1, . . . , n (1.2) and Aϕ(σ j ) = ρ j for j = 1, . . . , k, (1.3) where Aϕ(e iθ ) denotes the rate of change of the argument of ϕ(e iθ ) with respect to θ.
Problem 1.3 has been much studied, for example [39,40,11,29,26,42,27]. Without the tangential conditions (1.3), or some other constraint (for example, a degree constraint), the problem would arguably be ill-posed: solvability would depend only on the interpolation conditions at nodes in D, and the conditions at σ 1 , . . . , σ k would be irrelevant. With the conditions (1.3), however, the problem has an elegant solution. There is a simple criterion for the existence of a solution of Problem 1.3 in terms of an associated "Pick matrix", and better still, there is an explicit parametrization of all solutions ϕ by a linear fractional expression in terms of a parameter ζ ∈ T. There are polynomials a, b, c and d of degree at most n such that the general solution of Problem 1.3 is where the parameter ζ ranges over a cofinite subset of T (see Theorem 3.3 below). The polynomials a, b, c and d are unique subject to a certain normalization. Analogously, Problem 1.1 needs to be modified by the addition of tangential conditions at interpolation nodes in T in order to be well posed. We are led to the following refinement of Problem 1.1. Problem 1.4. Given Blaschke interpolation data (σ, η, ρ) with n interpolation nodes of which k lie in T, find if possible a rational Γ-inner function h = (s, p) of degree n such that h(σ j ) = (−2η j , η 2 j ) for j = 1, . . . , n and Ap(σ j ) = 2ρ j for j = 1, . . . , k.
We shall call this the royal Γ-interpolation problem with data (σ, η, ρ). The connection between Problems 1.4 and 1.3 can be described with the aid of a certain 1-parameter family of rational functions Φ ω on Γ, where ω ∈ T. These functions play a central role in the function theory of Γ (for example, [6,7]). They are defined by Φ ω is holomorphic on Γ, except for a singularity at (2ω,ω 2 ), and maps Γ into D − .  The Γ-inner functions whose range is contained in R, those of the form (2f, f 2 ) for some inner f , behave differently from others. Theorem 4.9 gives a formula for a solution h of Problem 1.4 in terms of s 0 , p 0 , a, b, c and d. Since the polynomials a, b, c and d are computed in Theorem 3.9 and Remark 3.11, we obtain an explicit solution of Problem 1.4. The algorithm is presented in Section 5.
The connection between the solution sets of the royal Γ-interpolation problem and the Blaschke interpolation problem can be made explicit with the aid of the functions Φ ω . Theorem 1.6. Let (σ, η, ρ) be Blaschke interpolation data. Suppose that h is a solution of Problem 1.4 with these data and that h(D) ⊂ R. For all ω ∈ T \ {−η 1 , −η 2 , . . . , −η k }, the function Φ ω • h is a solution of Problem 1.3 with the same data. Conversely, for every solution ϕ of the Blaschke interpolation problem with data (σ, η, ρ), there exists ω ∈ T such that ϕ = Φ ω • h.
This theorem is a corollary to Theorem 4.4.
In an earlier paper [3] the authors gave another construction of the general rational Γ-inner function h = (s, p) of degree n, starting from different data, to wit, the royal nodes of h and the zeros of s. One step in the construction in [3] is to perform a Fejér-Riesz factorization of a non-negative trigonometric polynomial, whereas, in contrast, the construction in this paper can be carried out entirely in rational arithmetic.

Background material
In this section we establish some notation and terminology and present some elementary facts about the set Γ discussed in the introduction.
The following results afford useful criteria for membership of Γ and bΓ [7].
will also arise. Proposition 2.1 implies that if h = (s, p) ∈ Hol(D, C 2 ) then h is Γ-inner if and only if p is inner, |s| is bounded by 2 on D and s(τ ) − s(τ )p(τ ) = 0 for almost all τ ∈ T with respect to Lebesgue measure (by Fatou's theorem, s and p have non-tangential limits a.e. on T). This paper focuses on the case that h is rational (that is, s and p are rational), in which case s =sp on the whole of T.
Let us clarify the notion of the degree of a rational Γ-inner function h.
is the homomorphism of fundamental groups induced by h when it is regarded as a continuous map from T to bΓ.
According to [3,Proposition 3.3], for any rational Γ-inner function h = (s, p), deg(h) is equal to the degree deg(p) (in the usual sense) of the finite Blaschke product p.
We denote by S the Schur class, which comprises all holomorphic maps from D to D − . Definition 2.3. For any differentiable function f : T → C \ {0} the phasar derivative of f at z = e iθ ∈ T is the derivative with respect to θ of the argument of f (e iθ ) at z; we denote it by Af (z).
Thus, if f (e iθ ) = R(θ)e ig(θ) is differentiable, where g(θ) ∈ R and R(θ) > 0, then g is differentiable on [0, 2π) and the phasar derivative of f at z = e iθ ∈ T is equal to The above is not standard notation, but we shall find it useful in the sequel. We summarise some elementary properties of phasar derivatives. (2) For any rational inner function ϕ and for all z ∈ T, 4) For any rational inner function p, Recall that a point λ ∈ D − is a royal node of a Γ-inner function h if and only if h(λ) is in the royal variety R = {(2z, z 2 ) : z ∈ C}.

The Blaschke interpolation problem and rational Γ-inner functions
The Blaschke interpolation problem, Problem 1.3, is an algebraic variant of the classical Pick interpolation problem. One seeks a Blaschke product of a given degree n satisfying n interpolation conditions, rather than merely a Schur-class function, and one admits interpolation nodes in both the open unit disc and the unit circle. As with the classical Nevanlinna-Pick problem, there is a criterion for the solvability of such a problem in terms of the positivity of a 'Pick matrix' formed from the interpolation data; however, to obtain a concise formulation, one has to impose additional interpolation conditions, on phasar derivatives at the interpolation nodes on the circle, and the bounds on these phasar derivatives appear on the diagonal entries of the Pick matrix. This modified Pick matrix appears in the work of several authors [10,4,40,25], but for simplicity we shall continue to speak of the Pick matrix. To be precise, the Pick matrix associated with Blaschke interpolation data (σ, η, ρ) as in Definition 1.2 is defined to be the n × n matrix M = [m ij ] n i,j=1 with entries Remark 3.1. Of course, it can happen for n-point Blaschke interpolation data (σ, η, ρ) that there exists a Blaschke product ϕ of degree strictly less than n satisfying the conditions (1.2) to (1.3), but in the present context we are concerned with solutions of degree exactly n.
In the case that n = k, that is, where all the interpolation nodes lie on the unit circle there is an elegant solvability criterion due to D. Sarason [40]. His result implies that, when n = k, Problem 1.3 is solvable if and only if the corresponding Pick matrix M is minimally positive, that is, when M ≥ 0 and there is no positive diagonal n × n matrix D, other than D = 0, such that M ≥ D. Actually, Sarason considers interpolation by functions in the Schur class, not just Blaschke products, and so there is a subtlety concerning the existence of phasar derivatives at boundary points (related to the Julia-Carathéodory theorem), but since we are only concerned with rational functions, no such difficulty will arise here.
The following result is well known -see [10,Sections 21.1 and 21.4] or [4,40,25]. Several authors have developed deep and far-reaching machines to characterise solvability of interpolation problems for classes related to Problem 1.3, and to parametrize their sets of solutions [10,11,19,30,20,25]; there is a brief history in [10, Notes for Part V, page 500]. A paper which addresses the combined interior and boundary problem specifically for finite Blaschke products is [26]. However, we have not found the precise statement that we need, and so, for the convenience of the reader, we give a self-contained treatment.
Our strategy for the construction of the general solution of Problem 1.3 is to adjoin an additional boundary interpolation condition; this augmented problem will have a unique solution, and in this way we obtain all solutions of Problem 1.3 in terms of a unimodular parameter.
The following is a refinement of the Sarason Interpolation Theorem [40], in that we consider interpolation nodes both on the circle and in the open disc. The result is contained in [18,Theorem 2.5]. See also [20,Theorem 5.2] for a solution to the analogous interpolation problem for the upper half plane. (1) There exists a function ϕ in the Schur class such that ϕ(σ j ) = η j for j = 1, . . . , n, (3.1) and the phasar derivative Aϕ(σ j ) exists and satisfies if and only if M ≥ 0; (2) if M is positive and of rank r < n then there is a unique function ϕ in the Schur class satisfying conditions (3.1) and (3.2), and this function is a Blaschke product of degree r; if and only if M is minimally positive.
Consider a point τ ∈ T distinct from σ 1 , . . . , σ k . For each ζ ∈ T we seek a solution ϕ to Problem 1.3 that satisfies the additional interpolation condition ϕ(τ ) = ζ and Aϕ(τ ) = ρ ζ,τ , where ρ ζ,τ > 0 is chosen to make the Pick matrix B ζ,τ of the augmented interpolation problem singular. We record the following simple lemma without proof. Lemma 3.4. If C is an n × n positive definite matrix, u is an n × 1 column, ρ = C −1 u, u and the (n + 1) × (n + 1) matrix B is defined by then B is positive semi-definite, rank(B) = n and The Pick matrix B ζ,τ of the augmented problem is the (n + 1) × (n + 1) matrix, where M is the Pick matrix associated with Problem 1.3, u ζ,τ is the n × 1 column matrix defined by and Thus the augmented problem that we are considering is the Blaschke interpolation problem with data (σ,η,ρ) wherẽ Proposition 3.5. Let ψ be a Blaschke product of degree N. Let σ = (σ 1 , σ 2 , . . . , σ n ) be an n-tuple of distinct points in D − , let η j = ψ(σ j ) for j = 1, . . . , n and let ρ j = Aψ(σ j ) for j such that |σ j | = 1. The Pick matrix for the data (σ, η, ρ) has rank at most N.
Proof. In the case that the σ j all lie in D the assertion is well known -see [4]. It follows easily from the fact that in this case the Pick matrix M is given by , where k λ denotes the Szegő kernel and T ψ is the analytic Toeplitz operator on the Hardy space H 2 with symbol ψ.
Proposition 3.6. If the Pick matrix M associated with Problem 1.3 is positive definite then, for any τ ∈ T \ {σ 1 , . . . , σ k } and ζ ∈ T there is at most one solution ϕ of Problem 1.3 for which ϕ(τ ) = ζ.
Proof. Let ψ be a solution of Problem 1.3 such that ψ(τ ) = ζ and let ρ τ = Aψ(τ ). Thus ψ is in the Schur class and satisfies ψ(σ j ) = η j for j = 1, . . . , n, Since ψ is a Blaschke product of degree n, it follows from Proposition 3.5, applied to the augmented problem with data (σ,η, (ρ, ρ τ )), that the corresponding Pick matrixM ρ τ has rank less or equal to n and so it is singular. Thus and so Aψ(τ ) is the same for every solution of Problem 1.3 such that ψ(τ ) = ζ. By Theorem 3.3, there is a unique function ψ in the Schur class satisfying the conditions (3.6), and hence there is at most one solution of Problem 1.3 such that ψ(τ ) = ζ.
We denote by e j the jth standard basis vector in C n . for j = 1, . . . , k, then there exists a unique solution ϕ to Problem 1.3 such that ϕ(τ ) = ζ.
In the light of Proposition 3.7 we define the exceptional set Z τ for Problem 1.3 to be Z τ = {ζ ∈ T : for some j, 1 ≤ j ≤ k, M −1 u ζ,τ , e j = 0}.
By the definition (3.8) and equation (3.10), Otherwise, Z j τ consists of at most one point ζ j τ ∈ T.
We shall call a point τ Suppose that every τ ∈ T \ {σ 1 , . . . , σ k } is unsuitable. Then Pick j 0 such that E j 0 is uncountable. Thus, for every τ ∈ E j 0 , By equations (3.9), Since the functions f i (λ) = 1 1−σ i λ , i = 1, . . . , n, restricted to the infinite bounded set E j 0 ⊂ C are linearly independent, c i = 0 for all i and M −1 e j 0 = 0. This is impossible. Therefore there exists τ ∈ T \ {σ 1 , . . . , σ k } such that the equalities Our final result concerning Problem 1.3 is that the particular solution guaranteed by Proposition 3.7 is uniquely determined by ζ and varies linear-fractionally in ζ. We suppose that Blaschke interpolation data (σ, η, ρ) are given, as in Definition 1.2. (1) If ζ ∈ T\Z τ , then there is a unique solution ϕ ζ of Problem 1.3 that satisfies ϕ ζ (τ ) = ζ. (2) There exist unique polynomials a τ , b τ , c τ , and d τ of degree at most n such that and, for all ζ ∈ T, if ϕ is a solution of Problem 1.3 such that ϕ(τ ) = ζ, then for all λ ∈ D.
and such that for three distinct points ζ in T \ Z τ , the equation holds for all λ ∈ D, then there exists a rational function X such thatã = Xa τ ,b = Xb τ ,c = Xc τ andd = Xd τ . Proof.
(1) By Proposition 3.8, there exists τ ∈ T \ {σ 1 , . . . , σ k } such that the set Z τ consists of at most k points. Proposition 3.7 asserts that if M is positive definite and ζ ∈ T\Z τ then there exists a solution ϕ to Problem 1.3 with ϕ(τ ) = ζ. By Proposition 3.6, the solution (when it exists) is unique.
To see the uniqueness of polynomials a τ , b τ , c τ , d τ assume that there is a second collection a 1 , b 1 , c 1 , d 1 of polynomials of degree ≤ n such that equations (3.13) and (3.14) hold. By what was proved in the previous paragraph, it is not the case that both collections of polynomials are relatively prime. Otherwise, there is a third collection a 2 , b 2 , c 2 , d 2 of polynomials of degree ≤ n − 1 such that equations (3.11) and (3.12) hold. This contradicts the fact that deg(ϕ) = n for all ζ ∈ T \ Z τ .
In view of Theorem 3.9 we can make precise what we mean by a parametrization of the solutions of a Blaschke interpolation problem. (2) for all but at most k values of ζ ∈ T, the function is a solution of Problem 1.3; (4) every solution ϕ of Problem 1.3 has the form (3.29) for some ζ ∈ T.
where the polynomials a τ , b τ , c τ and d τ are defined by equations (3.25). Note that different choices of τ will yield different normalised parametrizations.
In the terminology of Definition 3.10, Theorem 3.9 tells us the following. (1) at least one of the polynomials a, b, c, d has degree n; (2) the polynomials a, b, c, d have no common zero in C; (3) |c| ≤ |d| on D − .
Proof. As in Theorem 3.9 choose τ ∈ T \ {σ 1 , . . . , σ k } such that the set Z τ contains at most k points. Let the polynomials a = a τ , b = b τ , c = c τ and d = d τ be defined by equations (3.25). Theorem 3.9 shows that (a, b, c, d) has the properties (1), (2) and (3) of Definition 3.10. Let ϕ be a solution of Problem 1.3 and let ζ = ϕ(τ ). By Theorem 3.9(2), ϕ is given by equation (3.29). Hence property (4) of Definition 3.10 holds. Moreover (1) if all of a, b, c, d have degree strictly less than n then ϕ = aζ+b cζ+d is a rational function of degree strictly less than n, and so is not a solution of Problem 1.3.
(2) Suppose α ∈ C is a common zero of the polynomials a, b, c, d. On cancelling the common factor λ − α above and below in equations (3.29) and multiplying numerator and denominator by a suitable nonzero scalar we obtain a different normalized parametrization of solutions of Problem 1.3, with the same τ , contrary to the uniqueness statement in Theorem 3.9(2). Hence a, b, c and d have no common zero in C.
(3) By the normalization property in Definition 3.10 (3), Hence ad − bc is a polynomial of degree at most 2n and is not identically zero. Therefore Y def = {λ ∈ D : (ad − bc)(λ) = 0} contains at most 2n points.
Since c(τ ) = 0 and d(τ ) = 1, the continuous function f is strictly positive on a neighbourhood of τ in D. Suppose that f (λ 1 ) < 0 for some λ 1 ∈ D. Then f < 0 on an open set, and hence there are infinitely many points in D at which f = 0, a contradiction. Hence f ≥ 0 on D.

Prescribing the nodes and values
In this section we shall show how to construct rational Γ-inner functions with prescribed royal nodes and values. Our answer will be in terms of the solution to Problem 1.3 as described in Proposition 3.7 and Theorem 3.9. First we require a notion of multiplicity for royal nodes.
Definition 4.1. Let h be a rational Γ-inner function with royal polynomial R. If σ is a zero of R of order ℓ, we define the multiplicity #σ of σ (as a royal node of h) by The type of h is the ordered pair (n, k) where n is the sum of the multiplicities of the royal nodes of h that lie in D − and k is the sum of the multiplicities of the royal nodes of h that lie in T. We denote by R n,k the collection of rational Γ-inner functions h of type (n, k).

By [3, Theorem 3.8]
, if h = (s, p) belongs to R n,k then deg(h) = n and p is a Blaschke product of degree n.
The following example of rational Γ-inner functions from R n,k for even n ≥ 2 can be found in [1, Proposition 12.1].
Example 4.2. For all ν ≥ 0 and 0 < r < 1, the function belongs to R 2ν+2,2ν+1 . The royal nodes of h ν that lie in T, being the points at which |s| = 2, are the (2ν + 1)th roots of −1, that is, They are all of multiplicity 1. Note that there is a simple royal node at 0.
In this section we are concerned only with rational Γ-inner functions whose royal nodes all have multiplicity 1.
The next result provides a necessary condition for the existence of a rational Γ-inner function with prescribed royal interpolation data.
(2) Since Problem 1.3 is solvable, its Pick matrix is positive definite and so Theorem 3.9 tells us that there exist polynomials a, b, c, d of degree at most n which parametrise the solutions of Problem 1.3. Let us choose a particular such 4-tuple of polynomials, as described in Theorem 3.9. By Proposition 3.8, there exists τ ∈ T \ {σ 1 , . . . , σ k } such that the set Z τ (defined in equation (3.8)) consists of at most k points. Fix such a τ ∈ T; then there exist unique polynomials a τ , b τ , c τ , d τ of degree at most n such that and, for all ζ ∈ T \ Z τ , the function is the unique solution of Problem 1. for some rational function X. Let s 0 = s(τ ), p 0 = p(τ ). Since h is Γ-inner, equations (4.8) and (4.9) hold by virtue of Proposition 2.1. Since τ is chosen not to be a royal node of h, the inequation (4.10) also holds. Moreover |s 0 | < 2, since, for any point (z 1 , z 2 ) in the distinguished boundary bΓ of Γ, we have |z 1 | = 2 if and only if z 2 1 = 4z 2 -see [1, Proposition 3.2(3)]. It remains to prove equations (4.12) and (4.11).
There is a converse to Theorem 4.4. To prove it we need the following purely algebraic observation, which is proved by a routine calculation. Let rational functions s, p be defined by and let Then, as rational functions in (ω, λ), This algebraic relation has implications for rational maps from D to Γ. Suppose in addition that |p 0 | = 1, |s 0 | < 2 and s 0 =s 0 p 0 . Let rational functions s, p be defined by equations (4.34) and let (4.36) (i) If, for all but finitely many values of λ ∈ D, for all but finitely many ζ ∈ T, then s 0 c − 2d has no zeros in D − and (s, p) is a holomorphic map from D to Γ.
(ii) If, for all but finitely many ζ ∈ T, the function ψ ζ is inner, then h = (s, p) is a rational Γ-inner function.
Proof. (i) Notice first that the hypotheses on s 0 and p 0 imply that ζ(·) (defined by equation (4.35)) is an automorphism of D and so defines a bijective self-map of T.
By hypothesis there is a finite subset E of D such that, for all λ ∈ D \ E, there is a finite subset F λ of T such that the inequality (4.37) holds for all ζ ∈ T \ F λ .
We claim that the denominator s 0 c − 2d of s and p in equations (4.34) has no zeros in D − . For suppose that α is such a zero. Since |c| ≤ |d| on D − and |s 0 | < 2, at α, and hence d(α) = 0, and consequently c(α) = 0.
The following result gives the promised explicit construction of a solution of the royal Γ-interpolation problem in terms of a normalized parametrization of solutions of the corresponding Blaschke interpolation problem. Theorem 4.9. Let (σ, η, ρ) be Blaschke interpolation data with n distinct interpolation nodes of which k lie in T, as in Definition 1.2. Suppose that Problem 1.3 with these data is solvable and the solutions ϕ of Problem 1.3 have normalized parametrization Suppose that there exist scalars s 0 and p 0 such that and Then there exists a rational Γ-inner function h = (s, p) such that (i) h ∈ R n,k , (ii) h(σ j ) = (−2η j , η 2 j ) for j = 1, 2, . . . , n, (iii) Ap(σ j ) = 2ρ j for j = 1, 2, . . . , k. (iv) for all but finitely many ω ∈ T, the function Φ ω • h is a solution of Problem 1.3.
An explicit function h = (s, p) satisfying conditions (i)-(iv) is given by Proof. By Corollary 3.12 (3), |c| ≤ |d| on D − . Hence d(λ) c(λ) ≥ 1 for λ ∈ D − . By assumption |s 0 /2| < 1, and therefore s 0 c = 2d on D − . By Proposition 4.8, h is a rational Γ-inner function. Since a, b, c, d are polynomials of degree at most n, the rational function h has degree at most n. Recall that the degree of h coincides with the degree of p.
We have already observed that deg(h) ≤ n and that h(D) is not in R. Thus [3, Theorem 3.8] tells us that, in this case, the number of royal nodes of h is equal to the degree of h. Therefore h has at most n royal nodes. Since the points σ j , j = 1, 2, . . . , n are royal nodes, they comprise all the royal nodes of h and deg(h) = n. Precisely k of the σ j lie in T, and so h has exactly k royal nodes in T. Thus h ∈ R n,k and statement (i) holds.

The algorithm
In this section we summarize the steps in the solution of the royal Γ-interpolation problem in the form of a concrete algorithm.
We suppose given Blaschke interpolation data (σ, η, ρ) as in Definition 1.2. Here there are n prescribed royal nodes σ j , of which the first k lie in T and the remaining n − k are in D. To construct a rational Γ-inner function or functions of degree n having royal nodes σ j , royal values η j and phasar derivatives 2ρ j at σ j we proceed as follows.
(1) Form the Pick matrix M = [m ij ] n i,j=1 for the data (σ, η, ρ), with entries If M is not positive definite then the interpolation problem is not solvable. Otherwise, introduce the notation (where e j is the jth standard basis vector in C n ) is finite.
The following comments relate the steps of the algorithm to results in the paper. (1) If the royal Γ-interpolation problem is solvable, then the Blaschke interpolation problem with the same data is solvable, by Theorem 4.4. By Proposition 3.2, M > 0.
(2) This amounts to saying that Z τ is finite, in the notation of equation (3.8). By Proposition 3.8, there are uncountably many such τ ∈ T. The conditions that |s 0 | < 2, |p 0 | = 1 and s 0 =s 0 p 0 are equivalent to (s 0 , p 0 ) ∈ bΓ and |s 0 | < 2. By a standard parametrization of bΓ [7, Theorem 2.4], we can take s 0 = 2tω, p 0 = ω 2 for some t ∈ (−1, 1) and ω ∈ T. The condition (5.2) then becomes: for all λ ∈ D, After multiplication of both sides by n j=1 (1−σ j λ), the coefficients in this equation relating t and ω become polynomials in λ of degree at most n, and so the equation is in effect a system of 2n + 2 real equations in two real variables. Consequently the system is over-determined. The existence of s 0 , p 0 satisfying equations (5.2) is thus in principle a stringent condition for the solvablility of a royal Γ-interpolation problem. Remarkably, in the two examples in the next section, the λ terms factor out entirely from equation (5.3), and one obtains a single real equation in t and ω, which has a 1-parameter family of solutions. (4) The equations for a, b, c and d are equations (3.21) to (3.25). (5) The equations for s and p are (4.45) and (4.46).

Two examples
Even the simplest case of Problem 1.4, the royal Γ-interpolation problem with only one interpolation node, demands a surprising amount of calculation to solve. This problem is so simple that it can be readily solved without the foregoing theory, but it is instructive to see how the algorithm in Section 5 works in this case.
Example 6.1. Consider the case n = 1, k = 0 of Problem 1.4. There are prescribed a single royal node σ 1 ∈ D and a single royal value η ∈ D, and we seek a Γ-inner function h of degree 1 such that h(σ 1 ) = (−2η, η 2 ). By composition with an automorphism of D we may reduce to the case that σ 1 = 0. There is clearly at least a 1-parameter family of solutions, if any, since if h is a solution then so is h(ωλ) for any ω ∈ T.
The recipe for h in Section 4 proceeds as follows. Choose an arbitrary τ ∈ T. The normalized parametrization of the solution set of the associated Blaschke interpolation problem, according to equations (3.25), is given by The next step is to determine whether there exist s 0 , p 0 such that equations (4.8) to (4.13) hold. A little calculation shows that there is a 1-parameter family of such (s 0 , p 0 ), given by for any ω ∈ T. Substitution of these values into equations (4.45) and (4.46) yields the degree 1 Γ-inner function κ is a general point of T, and so we do obtain a 1-parameter family of Γ-inner functions of degree 1 satisfying h(0) = (−2η, η 2 ). An alternative expression for h is h(λ) = (β +βp(λ), p(λ)) where Example 6.2. Next consider the case of a single interpolation node on the unit circle -say σ = 1. A point η ∈ T and a ρ > 0 are prescribed, and we seek a Γ-inner function h = (s, p) of degree 1 such that h(1) = (−2η, η 2 ) and Ap(1) = 2ρ. Choose τ ∈ T \ {1}. Again calculate the normalized parametrization of the solution set of the associated Blaschke interpolation problem according to equations (3.25): Equations One can check directly that h = (s, p) is a Γ-inner function of degree 1 satisfying h(1) = (−2η, η 2 ) and Ap(1) = 2ρ. It appears at first sight that we have constructed a 2-parameter family of functions with the prescribed royal node, value and phasar derivative, since the parameters ω and τ range through T (or at least, cofinite subsets thereof). However, by means of some entertaining algebra, one can express h in terms of a single unimodular parameter (the same thing happened, though more simply, in Example 6.1). Let Clearly κ is unimodular. Now let It transpires that |α(κ)| < 1 and .
One may verify that the functions s and p in equations (6.1) can be written with κ ∈ T, evidently a 1-parameter family. It is noteworthy that the function h = (s, p) defined by equations (6.2) maps D into the disc {(η +ηz, z) : z ∈ D)}, which is a subset of the topological boundary ∂Γ of Γ. Inner functions h such that h(D) ⊂ ∂Γ were called superficial in [1] and discussed in [1,Proposition 8.3]. The example shows that the solutions of a royal Γ-interpolation problem can be superficial.

Concluding remarks
In this section we relate the results of the paper to some classical results in the theory of invariant distances and thereby describe some of the original motivation for our work.
The algorithm which is developed in this paper provides constructions of nextremal maps and m-geodesics in Hol(D, G) with prescribed royal nodes σ j , royal values η j and phasar derivatives at σ j . The n-extremal maps simultaneously generalize both Blaschke products and complex geodesics and constitute a significant class.
Recall that for a domain G in C N the Carathéodory distance C G on G is defined by C G (z 1 , z 2 ) = sup F ∈Hol(G,D) ρ(F (z 1 ), F (z 2 )). (7.1) In equation (7.1) z 1 and z 2 are two points in G, ρ denotes the pseudohyperbolic distance on D, ρ(z, w) = z − w 1 −wz .
and, for any domain G and any set E, Hol(G, E) denotes the space of holomorphic mappings from G to E. A dual notion is the Kobayashi distance of G, which is defined to be the largest pseudodistance K G subordinate to the Lempert function ρ G of G (e.g. [33,23]). The Lempert function of G is given by ρ G (z 1 , z 2 ) = inf h∈Hol(D,G) λ 1 ,λ 2 ∈D h(λ 1 )=z 1 h(λ 2 )=z 2 ρ(λ 1 , λ 2 ). (7. 2) The Kobayashi extremal problem for a pair of points z 1 , z 2 ∈ G is to find the quantity ρ G (z 1 , z 2 ) ( [31]). Any function h ∈ Hol(D, G) for which the infimum is attained is called a Kobayashi extremal function for the domain G and the points z 1 , z 2 . In the special case when G = G it turns out that the 1-parameter family Φ ω ∈ Hol(G, D), which we encountered in equation (1.5), is "universal" for the Carathéodory extremal problem [7,Corollary 3.4], the following sense. Theorem 7.1. If z 1 , z 2 ∈ G then there exists ω ∈ T such that C G (z 1 , z 2 ) = ρ(Φ ω (z 1 ), Φ ω (z 2 )). (7.3) Another fact about the complex geometry of G is that This corresponds to the geometric property of G that if h is an extremal function for the Kobayashi problem (7.2), then the range ran(h) of h is a totally geodesic analytic disc in G [7, Corollary 5.7]. The Kobayashi extremal problem can be viewed as an extremal 2-point interpolation problem. Specifically, by a finite interpolation problem in Hol(D, G), one means the following.
Problem 7.2. Given n distinct points λ 1 , . . . , λ n in D and n points z 1 , . . . , z n in an open or closed set G ⊂ C N , to determine whether there exists a function h ∈ Hol(D, G) such that h(λ j ) = z j for j = 1, . . . , n.
We say that Problem 7.2 is solvable, or that the data λ j → z j are solvable, if there does exist an h ∈ Hol(D, G) that satisfies these interpolation conditions. We say that the problem is (or the data are) extremal when the problem is solvable but there do not exist an open neighbourhood U of the closure of D and a map h ∈ Hol(U, G) such that the conditions h(λ j ) = z j for j = 1, . . . , n, (7.4) hold.
A map h ∈ Hol(D, G) is said to be n-extremal if, for any choice of n distinct points λ 1 , . . . , λ n ∈ D, the interpolation data λ j ∈ D → h(λ j ) ∈ G are extremally solvable.
With this perspective, if h and λ 1 , λ 2 minimize the right hand side of equation (7.2), then the 2-point interpolation problem λ j → z j , j = 1, 2 for Hol(D, G) is extremal and h is an extremal solution to it. Just as the Kobayashi extremal functions on G are both rational and Γ-inner, more generally, the following result obtains (see [22] or [2, Theorem 3.1]). Proposition 7.3. If λ j → (s j , p j ), j = 1, . . . , n, is a solvable n-point interpolation problem for Hol(D, G) then it has a rational Γ-inner solution.
The royal variety (or more precisely, R ∩ G) is a complex geodesic of G, with extremal function given by h R (λ) = (2λ, λ 2 ). Furthermore, among the complex geodesics in G, the royal variety is characterized by the property that α(R ∩ G) = R ∩ G whenever α is a biholomorphic self map (automorphism) of G [9, Lemma 4.3]. In addition, the automorphism group of G acts transitively on R ∩ G.
If F is a Carathéodory extremal function for some pair of points in G then so is m • F for any Möbius transformation m of the disc. The universal set described in Theorem 7.1 above is normalized so as to satisfy Φ ω • h R = −id D . As a result, Φ ω |R does not depend on ω.
(7.5) A Kobayashi extremal function on any domain for which the Lempert function and the Carathéodory distance coincide has a holomorphic left inverse. L. Kosinski and W. Zwonek [32] introduced a generalization of this notion: a map h : D → G, for any domain G, is said to be an n-complex geodesic if there exists a holomorphic map F : G → D such that F • h is a Blaschke product of degree at most n. The following result shows that rational Γ-inner functions enjoy this property.
Proposition 7.4. Let h be a rational Γ-inner function of degree n which is not superficial and let h(D) ⊂ R. Then (1) h is an (n+ 1)-extremal holomorphic map in Hol(D, Γ) and is an n-complex geodesic of G; (2) if in addition h has at least one royal node σ ∈ T then h is an n-extremal holomorphic map in Hol(D, Γ) and is an (n − 1)-complex geodesic of G.
Proof. First we recall a result from [8] that an analytic function h : D → G is a complex geodesic of G if and only if there is an ω ∈ T such that Φ ω • h ∈ Aut D and that every complex geodesic of G is Γ-inner. By Proposition 7.4, each nonsuperficial function from the set R 1,0 ∪ R 2,1 is a complex geodesic.