Poles of Painlev\'e IV Rationals and their Distribution

We study the distribution of singularities (poles and zeros) of rational solutions of the Painlev\'e IV equation by means of the isomonodromic deformation method. Singularities are expressed in terms of the roots of generalised Hermite $H_{m,n}$ and generalised Okamoto $Q_{m,n}$ polynomials. We show that roots of generalised Hermite and Okamoto polynomials are described by an inverse monodromy problem for an anharmonic oscillator of degree two. As a consequence they turn out to be classified by the monodromy representation of a class of meromorphic functions with a finite number of singularities introduced by Nevanlinna. We compute the asymptotic distribution of roots of the generalised Hermite polynomials in the asymptotic regime when $m$ is large and $n$ fixed.


Introduction
In this paper we address, by means of the isomonodromic deformation method [22,27,28], the distribution of movable singularities (which are zeros and poles) of rational solutions of the fourth Painlevé equation, also called Painlevé IV and denoted by P IV , which is the following second order differential equation, (1.1) particular classes of meromorphic functions (introduced by Nevanlinna [44]) with a finite number of critical points and transcendental singularities. One of the most striking features of solutions of Painlevé equations is that their value distributions are often observed to describe some approximate lattice structure, as was first discovered by Boutroux [3] for solutions of Painlevé I and II. This is also the case for singularities of Painlevé IV rationals. Indeed one of the main inspirations of our work is the highly regular pattern of their distribution, which was observed by Clarkson [8] (see [51] for more general solutions), and which has so far eluded any rigorous clarification. By computing the distribution of singularities in a particular asymptotic regime we furnish here the first rigorous, albeit partial, explanation 1 .
For the sake of clarity of our exposition, before introducing our main results together with an outline of the article, we briefly review some well-known facts from the theory of movable singularities and rational solutions of Painlevé IV.

Zeros and poles of solutions
The Painlevé property implies that any local solution of P IV has a unique meromorphic continuation to the entire complex plane [55]. As a consequence the solution space is the set of meromorphic functions on C that satisfy (1.1), which we denote by (

1.2)
Upon fixing an a ∈ C, any Painlevé IV transcendent (i.e., solution) enjoys a Laurent expansion at this point. In particular the generic Laurent expansion takes the form for b ∈ C * and c ∈ C, where the higher order coefficients are of the form ω n = 1 b p n , with p n polynomial in a, b, c, θ for n ≥ 2. However this expansion breaks down when ω(a) ∈ {0, ∞}, i.e., ω has a zero or pole at z = a, in which case the Laurent expansions take the respective forms ω(z) = 4 θ 0 (z − a) + b(z − a) 2 + ω 3 (z − a) 3 where we refer to the value of as the sign of the zero, or (1.5) where ω 1 = 1 3 (a 2 − 2 + 4(θ ∞ − )), in both cases b ∈ C, and all higher order coefficients have polynomial dependence on a, b, θ; conversely, for any choice of parameters the Laurent series (1.3)-(1.5) converge locally to a solution of P IV . Indeed, one can find a positive constant C a,b such that |ω n | ≤ C n a,b , for n ≥ 3, and thus the formal power-series actually has a non-zero radius of convergence, see, e.g., [21,26].
From an algebro-geometric point of view [29,48], it is clear that zeros and poles of P IV transcendents play a fundamentally distinguished role; they are the movable singularities of solutions.  Table 1. Action of Bäcklund transformations on rational solutions of P IV .

Rational solutions
Firstly, let us remark that P IV enjoys various Bäcklund transformations, which relate solutions with different parameter values. In particular we have transformations R 1 -R 4 , see Appendix B for their definitions, which allow us to relate the solution spaces W θ and W θ whenever θ − θ ∈ Z 2 ∪ (Z + 1 2 ) 2 . Painlevé IV admits a rational solution if and only if the parameters satisfy either Furthermore for any such parameter values the associated rational solution is unique [25,34,43]. Using the τ -function formalism, Noumi and Yamada [46] expressed these rational solutions conveniently in terms of generalised Hermite H m,n (z) and generalised Okamoto polynomials Q m,n (z), see Appendix A for their precise definition. Firstly, the parameter cases (1.6), up to equivalence θ 0 ↔ −θ 0 , are given by H m+1,n , θ 0 = 1 2 (m + n + 1), θ ∞ = 1 2 (n − m + 1), (1.8c) where m, n ∈ N, which we refer to as the Hermite I, II and III families respectively. Some particularly simple members of these respective families are and the other ones can be obtained via application of the Bäcklund transformations R 1 -R 4 , as depicted in Table 1. Secondly, the parameter cases (1.7), up to equivalence θ 0 ↔ −θ 0 , are given by ω m,n = − 2 3 z + d dz log Q m+1,n Q m,n , θ 0 = − 1 6 + 1 2 n, θ ∞ = 1 2 (2m + n + 1), (1.10) zeros ( = +1) zeros ( = −1) poles ( = +1) poles ( = −1)  Table 2. The zeros and poles of rational solutions in terms of generalised Hermite and Okamoto polynomials. As an example, poles with residue ε = +1 of ω (III) m,n coincide with the roots of H m,n+1 . The table is computed using equations (1.8) and (1.10)  where m, n ∈ Z, which we refer to as the Okamoto family. A particularly simple member of the Okamoto family, is given by and again the other ones can be obtained via application of the Bäcklund transformations R 1 -R 4 , as Table 1 shows.
Remarkably, all zeros and poles of rational solutions can be expressed as roots of the generalised Hermite and Okamoto polynomials, see Table 2. Therefore the study of the distribution of movable singularities of P IV rationals is reduced to that of zeros of the generalised Hermite and Okamoto polynomials.

Outline and main results
In Section 2 we recall the isomonodromic deformation interpretation of Painlevé IV, via the Garnier-Jimbo-Miwa Lax pair. The Riemann-Hilbert correspondence associates bijectively any solution of Painlevé IV to a unique monodromy datum of the Garnier-Jimbo-Miwa linear system; given a point z in the complex plane and a solution ω of P IV , the inverse monodromy problem furnishes the value ω(z) unless z is a zero or a pole, in which cases the inverse monodromy problem for the linear system does not have any solution.
However, following [37], we show that the inverse monodromy problem can be defined in case of zeros and poles, and in fact it simplifies to that of the anharmonic oscillator ψ (λ) = V (λ; a, b, θ)ψ(λ), V (λ; a, b, θ) = λ 2 + 2aλ + a 2 + 2 ( The main result of this section is that the aforementioned simplification allows us the characterise zeros and poles of rational solutions exactly as the solutions of an inverse monodromy problem concerning the anharmonic oscillator in question; see Theorem 2.2. According to the beautiful theory developed by Nevanlinna and his school [14,44], anharmonic oscillators (in case all singularities in the complex plane are Fuchsian and apparent) naturally define Riemann surfaces which are infinitely-sheeted coverings of the Riemann sphere uniformised by meromorphic functions. In Section 3 we define three families of such Nevanlinna functions and show how they classify the zeros and poles of rational solutions. This characterisation is rather powerful, as an easy corollary gives us the solution to a previously open problem, namely to determine exactly the number of real roots of the generalised Hermite polynomials; see Corollary 3.5. Finally, in Section 4 we study the asymptotic distribution of zeros of the generalised Hermite polynomials H m,n , and hence zeros and poles of corresponding rational solutions in the asymptotic regime m → ∞ and n bounded. In order to state precisely our main result we need to introduce some new notation and functions. First of all, as we will find that the roots grow like (2m + n) 1 2 , it is convenient in our analysis to work with the new unknown α and 'big parameter' E, defined by The m · n roots of H m,n (z) turn out to be organised in n approximately horizontal lines. We parametrise these lines by the set and introduce the real functions f : where j ∈ J n and the constant F n,j is defined by . (1.12) Notice that f is strictly monotone and therefore globally invertible. Finally, we define our approximate (rescaled) roots α.
Definition 1.1. For every k, with |k| ≤ E 4 and k integer if m is odd or half-integer if m is even, and every j ∈ J n , we define the approximate root α j,k = α R + iα I , with α R , α I ∈ R, as the unique complex number determined by the following relations The points α j,k are the dominant term in the asymptotic expansion of the (rescaled) roots belonging to the 'bulk', as it is pictorially illustrated in Fig. 1 below. Figure 1. Rescaled roots of H 40,5 . In blue numerically exact location, in red the asymptotic approximation α j,k . In purple the curves α = g j,E ( α), E = 85, j ∈ J 5 .
We consider two cases of 'bulk' behaviour. In the first case we compute the error in the approximation when we limit the real part of the roots to a fixed closed subinterval of (−1, 1).
There exist a constant E 0 and a constant C σ such that, if E ≥ E 0 , then for all |k| ≤ σE, the polynomial H m,n (z) has one and only one zero in each disc of the form z − E Remark 1.6. Our approximation scheme includes all zeros, except for a set which asymptotically has probability measure zero. Indeed let N (A, B) be the counting function for −∞ < A ≤ B < ∞ defined as follows: set N E (A, B) equal to the number of rescaled roots whose real part belong to the interval [A, B], divided by m = E−n 2 , and define N (A, B) = lim E→∞ N E (A, B). An easy computation, based on the identity f (x) = √ 1 − x 2 , shows that Here χ [−1,1] is the characteristic function of the interval [−1, 1]. This is a manifestation of Wigner's celebrated semi-circle law. This behaviour was conjectured in [20]. . These roots will be briefly discussed in Section 4.6 at the end of the paper where we show that their distribution is governed by an Airy-like behaviour.
Remark 1.8. While we were finalising the present paper, the paper [4] by R. Buckingham appeared on the arXiv. This contains the asymptotic analysis, in the regime m → ∞, m n = r, of rational solutions of Hermite type in the 'pole-free' region. Even though [4] does not address the distribution of poles and zeros, it contests the results of [47] about the distribution of singularities in the regime m = n 0. The paper [47] was however already peer-reviewed. Since we are not able to judge on this matter, we await for the answer of its authors.

Poles of rational solutions and anharmonic oscillators
It is well-known that Painlevé equations can be realised as isomonodromic deformations of linear systems. This section is dedicated to, using aforementioned formalism, giving an exact characterisation of poles of rational solutions in terms of anharmonic oscillators satisfying some prescribed constraints. Let us consider the following quadratic oscillators with centrifugal terms, where a, b ∈ C and θ = (θ 0 , θ ∞ ) ∈ C * × C. Without loss of generality, we may always assume [θ 0 ] ≥ 0, because of the invariance of (2.1) under θ 0 → −θ 0 . Now note that λ = 0 is a regular singular point with indices 1 2 ±θ 0 , and there generically exist two linearly independent solutions ψ ± (λ), with corresponding Frobenius expansions, uniquely defined by the asymptotic behaviour at zero. However, in the resonant case θ 0 = 1 2 n ∈ 1 2 N * , the Frobenius expansion of the dominant solution is not uniquely defined by its behaviour in zero ψ − (λ) ∼ λ 1 2 −θ 0 as λ → 0, and logarithmic terms may appear. Indeed, we have where the constant c and c k 's are uniquely defined by imposing c n = 0. For convenience of the reader we recall the definition of an apparent singularity, a concept of remarkable importance for the rest of the paper. Similarly, in the case of a first order system of linear ODEs, we say that a regular singularity is apparent, if the monodromy about the singularity is a scalar multiple of the identity.
In turn, equation (2.1) has an irregular singularity at λ = ∞, hence solutions exhibit the Stokes phenomenon, which we discuss briefly. For each k ∈ Z 4 , we define the Stokes sector In each Stokes sector there exists an (up to normalisation) unique subdominant solution ψ k (λ), which decays exponentially as λ → ∞ in this sector.
Fixing the principal branch for powers of λ in the λ-plane, with respect to the branch cut arg λ = − π 4 , and denoting we fix the following normalisation of the subdominant solution ψ k (λ), We write ψ ≡ φ, for solutions of (2.1), iff they are linearly dependent. We have the three following characterisations of roots of Hermite polynomials H m,n , i.e., zeros and poles of rational solutions of Hermite type, via an inverse monodromy problem for the anharmonic oscillator (2.1).
Furthermore, in such case, it turns out that ψ 0 (λ) ≡ ψ 2 (λ) and where p(λ) is a polynomial of degree m − 1 with no repeated roots and p(0) = 0, and g(λ, z) is as def ined in (2.2). This in particular implies that ψ 0 (λ) has exactly m − 1 nonzero simple roots in the complex plane.
Furthermore, in such case, it turns out ψ 1 (λ) ≡ ψ 3 (λ) and where p(λ) is a polynomial of degree n − 1 with no repeated roots and p(0) = 0, and g(λ, z) is as def ined in (2.2). This in particular implies that ψ 1 (λ) has exactly n − 1 nonzero simple roots in the complex plane.
Furthermore, in such case, it turns out that λ = 0 is an apparent singularity and where p(λ) and q(λ) are polynomials of degree n−1 and m−1 respectively without repeated roots and p(0), q(0) = 0, and g(λ, z) is as def ined in (2.2). This in particular implies that ψ 0 (λ) and ψ 1 (λ) have exactly n − 1 and m − 1 nonzero simple roots in the complex plane respectively.
To state the corresponding theorem for Okamoto rationals, we briefly discuss the Stokes multipliers for the anharmonic oscillator (2.1). If k = 1, 2, then the asymptotic characterisation (2.3) of the solution ψ k (λ) is in fact valid on the larger sector Ω k+1 ∪ Ω k ∪ Ω k−1 . It follows that {ψ k , ψ k−1 } is necessarily linearly independent and hence forms a basis of solutions of (2.1). Comparison of the asymptotic behaviour in Ω k as λ → ∞, of ψ k+1 and the basis elements, leads to the relation for a unique s k ∈ C, called the kth Stokes multiplier. Appropriately modifying the above argument for the k = 0, 3 cases, due to the choice of branch in the λ-plane, leads to for unique s 0 , s 3 ∈ C.
We have the following characterisation of roots of the generalised Okamoto polynomials, or equivalently of zeros and poles of rational solutions of Okamoto type. Theorem 2.3. Fix m, n ∈ Z, then z = a is a zero of the generalised Okamoto polynomial Q m,n (z), if and only if there exists b ∈ C, such that the Stokes data of the anharmonic oscillator (2.1) with θ 0 = 1 2 n − 1 6 and θ ∞ = 1 2 (2m + n + 1), satisfy s 0 = s 2 , s 1 = s 3 , s 0 s 1 + 1 = 0.
The remainder of this section is dedicated to the proofs of Theorems 2.2 and 2.3. In Section 2.1 we discuss the realisation of P IV as an isomonodromic deformation. Then, in Section 2.2 we show that, upon localising the isomonodromic system near poles, we arrive at the anharmonic oscillator (2.1), while preserving the monodromy of the system. This allows us to characterise poles of P IV transcendents as solutions of certain inverse monodromy problems concerning the anharmonic oscillator, in particular leading to proofs of the aforementioned theorems in Section 2.3.
Remark 2.4. In recent literature some special interest is given to anharmonic oscillators for which one solution is expressible in closed form as a polynomial (or rational function) times an exponential function. These are called quasi exactly solvable potentials [1,56]. We remark here that, by Theorem 2.2 (see [35] for some similar considerations), all three types of oscillators related to the roots of generalised Hermite polynomials are quasi exactly solvable in this sense. Moreover, the type III case is particularly special as all solutions are expressible in terms of a linear combination of rational functions times an exponential and hence deserve to be called super quasi exactly solvable. It is interesting to note that, while quasi exactly solvable potentials are mostly studied with the help of the sl 2 Lie algebra, in the present case they are related to Bäcklund transformations of P IV , which form the affine Weyl group of type A

The Garnier-Jimbo-Miwa Lax pair
In this section, we introduce the isomonodromic deformation method for P IV , as developed by Kitaev [32], Ablowitz et al. [24] and Kapaev [30,31]. The main aim of this section, is to characterise rational solutions by means of monodromy data of an associated linear problem.
Jimbo and Miwa [28] realised P IV as the compatibility condition of the following two linear differential systems where u is an auxiliary function satisfying and v is defined by That is, Y λz = Y zλ implies that ω and u satisfy P IV (θ) and (2.7) respectively. Note that the linear system (2.6a), associated to P IV , is not defined at zeros and poles of ω. To study what happens near such points, we follow the strategy pioneered by A. Its and V. Novoskhenov for Painlevé II [27], and then employed by one of the authors [36,37] for Painlevé I, see also [7]. It turns out that, even though the matrix linear system associated to P IV is not defined at zeros and poles, on the contrary its scalar version has a well-defined limit. We consider the equivalent 2nd order scalar differential equation for the gauged first component which we refer to as the scalar Garnier-Jimbo-Miwa (GJM) Lax pair, where Besides λ = 0 regular singular and λ = ∞ an irregular singular point, equation (2.8a) has a further apparent singularity at λ = 1 2 ω, with indices {0, 2}. So solutions of (2.8a) can branch only at λ = 0 and λ = ∞, and hence live on the universal covering space of C * , which we denote by C ∞ . Lemma 2.5. Any local solution Ψ(λ, z) of (2.8), extends uniquely to a global single-valued meromorphic function on C ∞ × C, singular in z only where ω(z) has a pole with +1 residue, in which case Ψ(λ, z) has a simple pole in z. The behaviour of Ψ(λ, z) near zeros and poles of ω(z) is characterised as follows: • If ω(z) has a zero with negative sign at z = a, say corresponding expansion takes the form (1.5) with = −1, then Ψ(λ, z) is holomorphic at z = a and ψ(λ) := λ − 1 2 Ψ(λ, a) defines a solution of the anharmonic oscillator ψ (λ) = λ 2 + 2aλ + a 2 + 1 − 2θ ∞ + 1 2 bλ −1 + θ 0 (θ 0 + 1)λ −2 ψ(λ).
• If ω(z) has a zero with positive sign at z = a, say corresponding expansion takes the form (1.5) with = 1, then Ψ(λ, z) is holomorphic at z = a and ψ(λ) := λ − 1 2 Ψ(λ, a) defines a solution of the anharmonic oscillator • If ω(z) has a pole with residue −1 at z = a, say corresponding expansion takes the form (1.5) with = −1, then Ψ(λ, z) is holomorphic at z = a and ψ(λ) := Ψ(λ, a) defines a solution of the anharmonic oscillator (2.1), namely • If ω(z) has a pole with residue +1 at z = a, say corresponding expansion takes the form (1.5) with = +1, then Ψ(λ, z) has a simple pole at z = a, and corresponding residue ψ(λ) := lim z→a (z − a)Ψ(λ, z) defines a solution of the anharmonic oscillator Proof . Note that, without assuming the Painlevé property of P IV , the statement of the lemma is rather nontrivial, as it in particular implies the Painlevé property for P IV , see for instance Fokas et al. [22]. However, we know that, around any point z = a in the finite complex plane, ω(z) has a convergent Laurent expansion, taking any of the forms (1.3)-(1.5), which greatly simplifies the proof. The unique global single-valued meromorphic continuation of local solutions is proved by showing: 1) near any point (λ, z) = (λ 0 , a) ∈ C ∞ × C, there exists a meromorphic local fundamental solution 2 of (2.8); 2) any two local fundamental solutions Y (λ, z) and Y (λ, z) of (2.8), with non-empty intersection of domains, are related by a constant invertible matrix C as in (2.17), on this intersection.
To prove the first statement, we have to distinguishing between the different possibilities (1.3)-(1.5), and whether λ 0 = 1 2 ω(a) or not. We give a detailed account of the case where z = a is a pole with residue −1 of ω(z). The other cases are proven analogously.
So suppose z = a is a pole with residue −1 of ω(z), and let b ∈ C be defined by (1.5) with = −1. Firstly, we choose small open discs λ 0 ∈ Λ ⊆ C ∞ and a ∈ Z ⊆ C such that ω(z) has no zeros or poles on Z \ {a} and 2λ − w(z) does not vanish on Λ × (Z \ {a}).
Writing U 1 = Ψ and U 2 = Ψ λ , equations (2.8) can be rewritten as Furthermore, note that, as ω solves P IV , the compatibility condition is satisfied. Direct calculation gives locally uniformly in λ ∈ Λ, where V (λ; a, b, θ) is the potential defined in (2.1). Using this fact, it easily follows that A(λ, z) is analytic on Λ × Z.
We will show the existence of a unique fundamental solution U(λ, z) of equations (2.9), analytic on Λ × Z, with U(λ 0 , a) = I. To this end, firstly note that the Cauchy problem We now search for a solution of (2.9), of the form where F (z) analytic on Z with F (a) = I, independent of λ. Note that U(λ, z) trivially satisfies (2.9a), and (2.9b) reduces to Hence, to show that the Cauchy problem (2.14) has a unique solution F , independent of λ, it remains to show that C λ = 0. By direct calculation, using where the latter equality is precisely equation (2.10). We conclude that the Cauchy problem (2.9b) has a unique fundamental solution F (z), analytic on Z, which is independent of λ. Equation (2.13) now defines a unique fundamental solution U(λ, z) of (2.9), analytic on Λ × Z, with U(λ 0 , a) = I. Set then Y * (λ, z) defines a fundamental solution of (2.8), on the open environment Λ × Z of (λ 0 , a). Furthermore, note that, by equation (2.11), with the potential V (λ; a, b, θ) as defined in (2.1). In particular Y * (λ, a) denotes a local fundamental solution on Λ of the anharmonic oscillator (2.1). Considering part (2), if Y (λ, z) and Y (λ, z) are local fundamental solutions, with non-empty intersection of domains, then there exists a unique meromorphic matrix C = C(λ, z) specified by equation (2.17) on this intersection. By equations (2.8a) and (2.8b), we have respectively C λ = 0 and C z = 0, hence C ∈ GL 2 (C) is constant on this intersection.

16)
holds at z = z * , then (2.16) holds globally in z. Furthermore, suppose Y (λ, z) and Y (λ, z) are fundamental solutions of (2.8), then there exists a constant matrix C ∈ GL 2 (C) such that (2.17) Proof . Using the deformation equation (2.8b), one inductively shows that Hence (2.16) holds locally in z around z = z * , and hence globally by Lemma 2.5. The second part follows from item (2) in the proof of Lemma 2.5.
We write Ψ ≡ Φ, for solutions of (2.8), iff they are linearly dependent. Ablowitz et al. [24] show that, for k ∈ Z 4 , there exists a unique subdominant solution of (2.8) in Ω k , normalised by means of the asymptotic characterisation as λ → ∞ in Ω k , locally uniformly in z away from zeros and poles of ω(z). The Stokes phenomenon near λ = ∞ in (2.8) now translates to the exponentially small jumps for a unique s k = s k (z) ∈ C, called the kth Stokes multiplier. From Lemma 2.6, we immediately obtain that the Stokes multipliers s k = s k (z) are constant with respect to z. Considering (2.8) near λ = 0, we restrict our discussion to θ 0 / ∈ − 1 2 N. Kapaev [30,31] shows the existence of solutions Ψ 0 1 (λ, z) and Ψ 0 2 (λ, z) of (2.8) with Frobenius expansions in λ, at λ = 0, is independent of z, by Lemma 2.6. We call λ = 0 an apparent singularity if j = 0. The main objective of this section, is to prove the following two propositions. in Ω 0 , is also subdominant at λ = 0, i.e., Ψ 0 (λ, z) ≡ Ψ + (λ, z).
Proposition 2.9. Fix m, n ∈ Z, let θ 0 = 1 2 n − 1 6 and θ ∞ = 1 2 (2m + n + 1), then a solution ω ∈ W θ equals the rational solution w m,n (z), if and only if the Stokes data of the scalar GJM Lax pair (2.8) satisfy In order to prove Propositions 2.7 and 2.9, we first discuss the monodromy space and monodromy mapping induced by the scalar GJM Lax pair. There are different cases to be taken care of, the non-resonant and resonant, both of which are necessary to study rational solutions. We then discuss the monodromy corresponding to the rational solutions. The Okamoto and Hermite I and II families of rational solutions, can be found in the literature [31], see also [42]. However, the Hermite III case seems to be missing in the literature; hence we study it in Proposition 2.14 below. Finally we use these data to prove aforementioned propositions. For sake of later convenience, we introduce the following matrices

Monodromy of linear problem
We discuss the monodromy data of the scalar GJM Lax pair, following Ablowitz et al. [24] and Kapaev [30,31] closely. Let us define fundamental solutions of the scalar GJM Lax pair (2.8) near λ = ∞, and a fundamental solution near λ = 0, Given any solutions Ψ and Φ of the scalar GJM Lax pair (2.8), it follows by direct calculation that there exists a c ∈ C such that their Wronskian equals It is easily seen that Y 0 (λ, z) and Y 0 (λ, z) have identical Wronskian, given by c = 1 in the above formula. Therefore, by (2.23), the connection matrix E has unit determinant. For k ∈ Z 4 , we call S k the k-th Stokes matrix, which, by (2.19), equals which, by taking traces, implies as M 0 is given explicitly by Now the Stokes data s = (s 0 , s 1 , s 2 , s 3 ) and connection matrix E depend on the choice of solution ω ∈ W θ , and auxiliary functions u and κ, characterised by (2.7) and (2.21) respectively. Different choices u = αu and κ = βκ, with α, β ∈ C * , lead to a change of monodromy data given by Let M s θ denote space obtained by cutting s ∈ C 4 with respect to (2.26).
Definition. In the non-resonant case θ 0 / ∈ 1 2 Z, the monodromy space M θ is defined as the quotient of M s θ with respect to the action given in (2.28). We denote the corresponding monodromy mapping by which sends a solution of P IV (θ) to corresponding orbit of Stokes data of (2.8).
In the resonant case θ 0 ∈ 1 2 N * , the dominant solution Ψ − (λ, z) at λ = 0, is no longer uniquely specified by the asymptotic expansion in (2.20b) as one can add arbitrary multiples of the subdominant solution Ψ + (λ, z) to it. This amounts to arbitrary right multiplication of the fundamental solution at λ = 0, which correspondingly transforms E as E = (I + rσ + )E. Invariant under this transformation, and the action induced by changing κ in (2.29), is the quantity Lemma 2.11. Let θ 0 ∈ 1 2 N * , then λ = 0 is an apparent singularity of (2.6a), i.e., j = 0, if and only if the stokes multipliers s are elements of the one-dimensional submanifold M 0 θ of M s θ , defined by Furthermore, in case j = 0, then the quantity i E is given explicitly by where we note that numerator and denominator of the right-hand side vanish simultaneously if and only if (2.31) holds, for s ∈ M s θ . Proof . This follows directly from equations (2.25) and (2.27), see also Kapaev [31].
Note that the action in (2.29) gives and we define In the resonant case θ 0 ∈ 1 2 N * , the monodromy space M θ is defined as the quotient of M θ with respect to the action given in (2.28) and (2.32). We denote corresponding monodromy mapping by which sends a solution of P IV (θ) to corresponding orbit of monodromy data of (2.8).
Finally, the monodromy data, are not only invariant under the P IV flow, but also under the action of the Bäcklund transformations R 1 -R 4 , defined in Appendix B.

Monodromy corresponding to rational solutions
Proposition 2.14. The monodromy data corresponding to the Hermite I family (1.8a), are given by The monodromy data corresponding to the Hermite II family (1.8b), are given by The monodromy data corresponding to the Hermite III family (1.8c), are given by The Stokes data corresponding to the Okamoto family (1.10), are given by Proof . As to the Okamoto family, see Kapaev [31] and Milne et al. [42]. In the former paper, Kapaev also handles the Hermite I and II cases. Let us consider the Hermite III case. Because the monodromy data are invariant under Bäcklund transformations R 1 -R 4 , by Proposition 2.13, we only consider the simple case (1.9c). Then, for any α ∈ C * , a solution of (2.7) is given by u(z) = α. Now, considering the scalar GJM Lax pair (2.8), it follows by direct calculation that a fundamental solution is given by Then, comparison with the asymptotic characterisations (2.18), gives which implies that all Stokes multipliers vanish and λ = 0 is an apparent singularity. Now, for any β ∈ C * , a solution of (2.21) is given by κ(z) = βz. It is straightforward to check that Proof of Proposition 2.7. Let us consider the Hermite I case H.I. Because of the injectivity of the monodromy mapping 2.12, to establish the first part, all we have to show is that the monodromy data corresponding to Hermite I (2.33), are equivalent to (2.8) having an apparent singularity at λ = 0, and Ψ 0 (λ, z) ≡ Ψ + (λ, z). Now suppose that the monodromy data of (2.8) are given by (2.33), then Lemma 2.11 shows that λ = 0 is indeed an apparent singularity. The fact that i E = 0 readily translates to Ψ 0 (λ, z) ≡ Ψ + (λ, z). Conversely, suppose λ = 0 is an apparent singularity and Ψ 0 (λ, z) ≡ Ψ + (λ, z), then the latter immediately gives i E = 0. Furthermore Lemma 2.11 shows Hence s 1 = 0 or s 2 = 0. Suppose, for the sake of contradiction, that s 2 = 0. Then Ψ 1 (λ, z) = Ψ 3 (λ, z) and hence It is now easily seen that is an entire function satisfying f (λ, z) ∼ λ −m−1 as λ → ∞ in C. Such a function does not exist, hence s 2 = 0 and we are left with s 1 = s 3 = 0.
Considering the second part of the Hermite I case H.I, as s 1 = 0, we indeed have Ψ 0 (λ, z) ≡ Ψ 2 (λ, z). Let us define P 1 (λ, z) by equation (2.22). Then it is easily seen that P 1 (λ, z) is entire satisfying P 1 (λ, z) ∼ − 1 2 u(z)λ m , as λ → ∞ in C, from which it follows that P 1 (λ, z) is a polynomial in λ of degree m. The expression for the constant term of P 1 (λ, z) stems from the asymptotic characterisation of Ψ + (λ, z). The cases H.II and H.III follow by a similar line of argument.

Localisation of Lax pair at poles
Recall that the scalar GJM Lax pair (2.8) has a regular singular point at λ = 0, an irregular singular point at λ = ∞ and a further apparent singularity at λ = 1 2 ω. Now, considering Lemma 2.5, a pole of ω with −1 residue is a point where the further apparent singularity merges with the irregular singular point, resulting in an integer jump of one of the exponents of the irregular singular point, as can be seen by comparison of the asymptotic expansions (2.18) and (2.3), in the k is odd case. In this section, we wish to show that the monodromy of the scalar GJM Lax pair is preserved in such a limit.

Monodromy of anharmonic oscillator
Let us reconsider the anharmonic oscillator (2.1), for some fixed a, b ∈ C. For k ∈ Z 4 , we defined unique solutions ψ k (λ), subdominant in Ω k , by (2.3). Let us also recall the Stokes phenomenon of the anharmonic oscillator near λ = ∞, made explicit by equations (2.4) and (2.5), with corresponding Stokes data s ∈ C 4 . Similar to equations (2.20), there exist, for θ 0 / ∈ − 1 2 N, solutions of (2.1) enjoying Frobenius expansions near λ = 0 of the form where f 1 (λ) and f 2 (λ) entire and j = 0 in the non-resonant case θ 0 / ∈ 1 2 N * . There exists a unique matrix E ∈ GL 2 (C), which we call the connection matrix, such that As both fundamental solutions appearing in the above equation have unit Wronskian, the connection matrix has unit determinant.
Let us define S k by (2.24), define M 0 by (2.27), then equations (2.25) and hence (2.26) hold, so s ∈ M s θ . Furthermore Lemma 2.11 also holds true in this case. We define i E ∈ P 1 by (2.30) if θ 0 ∈ 1 2 N * . Finally we define the monodromy mapping for the anharmonic oscillator (2.1), ) is the orbit corresponding to the monodromy data {s, E} of (2.1) within M θ .

Localisation and Monodromy
We now wish to compare the monodromy data of the GJM scalar equation (2.8) and corresponding anharmonic oscillator (2.1), upon localisation.
Lemma 2.15. Let ω(z) be a solution of P IV and fix some u(z) and κ(z) satisfying (2.7) and (2.21) respectively. Say z = a is a pole of ω(z) with residue −1 and corresponding coefficient b as in (1.5) with = −1, and let u 0 , κ 0 = 0 be defined by Then, for k ∈ Z, we have
We define the Laurent mapping where L ± θ (a, b) denotes the meromorphic continuation of (1.5) with = ±1. Proposition 2.16. For any parameter values θ ∈ C * × C, where T θ is the monodromy mapping of the scalar GJM Lax pair, T o θ is the monodromy mapping of the anharmonic oscillator and L − θ is the Laurent mapping (2.37). Proof . Take any (a, b) ∈ C 2 , then ω := L − θ (a, b) is a solution of P IV (θ) with Laurent expansion (1.5) about z = a with = −1. Let us take some u(z) and κ(z) satisfying (2.7) and (2.21) respectively, which we may assume are normalised such that u 0 = κ 0 = 1 in (2.34). Then, using Lemma 2.15, it is easy to see that the the Stokes multipliers, and quantity i E in the resonant case, are conserved as z → a. Proof . We have already established the "only if" part. As to its converse, suppose b ∈ C is such that T o θ (a, b) = T θ (ω). Let ω be the solution of P IV (θ) defined by ω = L − θ (a, b). Then we know, by Proposition 2.16, that T θ ( ω) = T o θ (a, b) = T θ (ω). Hence we have ω = ω, by the injectivity of the monodromy mapping, see Propositions 2.10 and 2.12. In particular z = a is indeed a pole with residue −1 of ω, and b is the coefficient in (1.5).

Exact characterisation of poles
Note that Theorem 2.3 is a direct consequence of Proposition 2.14 and Theorem 2.17. However, Theorem 2.2 still requires some work.
Proof of Theorem 2.2 and Remark 2.8. We fix m, n ∈ N * and let us consider the equivalence with H.1. We set θ 0 = 1 2 n and θ ∞ = m + 1 + 1 2 n. From Proposition 2.14 and Theorem 2.17, we conclude that, z = a is a zero of H m,n (z), if and only if, there exists b ∈ C such that the monodromy of the anharmonic oscillator (2.1) is given by (2.33). The latter statement is easily seen to be equivalent to H.1, by an argument identical to the proof of the first part of H.I in Proposition 2.7. Furthermore, in case z = a is indeed a zero of H m,n (z), then comparison of H.1 and H.I, gives P 1 (λ, a) = 1 2 u 0 p 1 (λ), by Lemma 2.15. The roots of p 1 (λ) are necessarily simple. Now P 1 (λ, z) might a priori have a double root at λ = 1 2 ω(z), for special values of z. However, by Lemma 2.6, this would imply that P 1 (λ, z) has a double root at λ = 1 2 ω(z), for all values of z, not equal to a zero or pole of ω(z). Indeed, the latter follows from the fact that there exists an up to scalar multiplication unique solution of (2.8), which has a double root at λ = 1 2 ω(z), for all values of z, not equal to a zero or pole of ω(z).
Because of the identity P 1 (λ, a) = 1 2 u 0 p 1 (λ), this would in turn imply that p 1 (λ) is of degree at most m − 2, in contradiction with the fact that p 1 (λ) is necessarily of degree m − 1. We conclude that P 1 (λ, z), does not has a double root at λ = 1 2 ω(z), for all values of z, not a zero or pole of ω(z). In particular, for any such z, all the roots of P 1 (λ, z) are simple and nonzero.
The equivalence with H.2 and H.3 is shown by the same line of argument.

Nevanlinna functions and poles of rational solutions
We have showed that poles of rational solutions are in bijection with anharmonic oscillators having particular prescribed monodromy. Here we show that these oscillators naturally define Riemann surfaces which are infinitely-sheeted coverings of the Riemann sphere uniformised by meromorphic functions. More precisely we introduce two discrete classes of Riemann surfaces and we show that they classify roots of the generalised Hermite polynomials. We then derive a slightly weaker characterisation for roots of Okamoto polynomials.
Our approach is based on the seminal work by Nevanlinna [44] and Elfving [14], which has lately been revived and found useful in modern applications, see, e.g., [18,40].
The following Definition is instrumental to the analysis below.
Definition 3.1. Let K be a one dimensional complex manifold.
A holomorphic function f : K → P 1 (that is a meromorphic function) is called a branched covering of the sphere if there exists a finite subset S ⊂ P 1 such that the restriction The minimal set B among all the sets satisfying the above property is called the branching locus of f . We remark that in what follows we will always restrict to the case when K is either the complex plane or the Riemann sphere.

Anharmonic oscillators and Nevanlinna theory
Before tackling our characterisation, we briefly sketch the theory of the Riemann surfaces associated with anharmonic oscillators and suggest [40] for a complete introduction. Consider an anharmonic oscillator where r is some rational function, with at most double poles in the complex plane. Then, for any two linearly independent solutions {ψ, φ} of (3.1), the function f = ψ/φ : C → P 1 is locally invertible at any point λ, unless λ is a double pole of r. Suppose λ * is a double pole and locally r(λ) = n 2 −1 4(λ−λ * ) 2 + O( 1 λ−λ * ) for some n ∈ C. Then the corresponding indices of (3.1) are 1±n 2 . If n ∈ N and the Fuchsian singularity is apparent, then f is locally single valued, but in this case λ * is a critical point of f of order n − 1. If, on the contrary, n ∈ N but the Fuchsian singularity λ * is not apparent, or n / ∈ N, then f is a multivalued function in the neighbourhood of λ * .
The point at infinity is in general an irregular singularity. Suppose r(λ) = λ M + O λ M −1 as λ → ∞, then equation (3.1) admits M + 2 Stokes sectors Each Stokes sector can be thought as a critical point of infinite order -technically a logarithmic direct transcendental singularity [17]. Indeed, in each Stokes sector, the function f has a welldefined asymptotic value while all its derivatives vanish exponentially fast; here the limit must be taken along curves not tangential to the boundary of the Stokes sector. These asymptotic values can be directly computed from the Stokes multipliers but we do not need the general relation here [39,44]. Now, f is only unique up to composition by Möbius transforms. Indeed if {ψ * , φ * } is another choice of linearly independent solutions of (3.1), then Summing up, to any anharmonic oscillator (3.1), such that all poles of the potential r in the plane are apparent Fuchsian singularities, we can associate a branched covering of the sphere, up to automorphism of the target sphere, namely up to Möbius equivalence. The branching locus B is the union of the critical values and asymptotic values.
In turn, one can recover the potential r from f , by means of the Schwarzian derivative,

which indeed is invariant under composition by Möbius transformations (3.2).
Finally, we notice that two meromorphic functions f and f * , of the kind described above, are topologically equivalent coverings of the sphere if and only if there exist c = 0 and d such that f * (λ) = f (cλ + d).
The question that remains is whether all branched coverings of the sphere as above can be obtained by means of anharmonic oscillators. This was positively settled by Elfving, as the following theorem shows. Proof . For the case m = 0 see [44]. For the general case, see [14].

Combinatorics of branched coverings
In order to present our results, we briefly introduce the concept of a line complex corresponding to a branched covering of the sphere, and refer the reader to [14,40,44] for the precise definitions. Let f be a branched covering of the sphere, with ordered branching locus {b 1 , . . . , b n }, and let us fix an oriented Jordan curve γ, passing through all of the branching points, respecting the particular ordering. This curve divides the sphere into an inner and outer polygon, with common sides given by the arcs (b 1 , b 2 ), . . . , (b n , b 1 ), see Fig. 3.
We choose a point P i in the inner polygon and a point P o in the outer polygon, as well as for each 1 ≤ k ≤ n, an analytic line (i.e., curve) l k connecting P i and P o , going only through the side (b k , b k+1 ) and only once, with convention b n+1 = b 1 . The line complex is the graph, given by the inverse image under f of the union of lines l k , where the set of vertices V is given by the (disjoint) union of V i := f −1 (P i ) and  Critical points of f are encircled by a closed loop of the graph with more than one internal (and external) vertex; the multiplicity of the critical point is the number of internal or external vertices in the loop minus one. If b k is the corresponding critical value, then the monodromy representation σ k acts on the internal vertices of the loop by a simple shift.
Stokes sectors are bounded by an infinite sequence of internal and external vertices. If b k is the corresponding critical value, then the monodromy representation σ k acts transitively on the internal vertices of the sequences of internal vertices by a simple shift.
Note that the line complex of f , up to orientation-preserving homeomorphism of the domain, is far from unique, as it depends on the particular ordering of branch points and of the curve γ. However, for fixed choice of the ordering and curve, to any line complex corresponds a unique equivalence class of meromorphic functions. Moreover, as to be expected, the braid group acts transitively on the possible line complexes of f [14, 33].

Hermite oscillators and families of coverings
In this subsection we construct two distinguished families of branched coverings of the sphere and show that they classify roots of generalised Hermite polynomials.
Definition. For m, n ∈ N * , we call the anharmonic oscillator (2.1) (or its potential) with θ 0 = 1 2 n and θ ∞ = m + 1 +  Notice that all functions in these families are Belyi functions [40] because the five singularities lie over three distinct points.
In Fig. 4 (resp. 6), where we use the notation depicted in Fig. 5, we classify all line complexes describing functions in the families F 1 (resp. F 2 ), according to the combinatorics of coverings described above. Each line complex is completely determined by a set of arbitrary positive integers n 1 , . . . , n 4 . In Fig. 4, the values of b 1 , b 2 , b 3 are b 1 = −i, b 2 = 0 and b 3 = i and the curve γ is chosen to be the imaginary line. In Fig. 6, the values of b 1 , b 2 , b 3 are b 1 = 1, b 2 = 0 and b 3 = ∞ and the curve γ is chosen to be the real axis.
By the general theory, Fig. 4 (resp. 6) classifies all functions in F 1 (resp. F 2 ). Indeed each of the line complexes described determines a unique function in F 1 (resp. F 2 ), and vice versa for every f ∈ F 1 (resp. f ∈ F 2 ) the line complex of f is one of those depicted in Fig. 4 (resp. 6).
One can read from the line complex in Fig. 4 that the critical point and two transcendental singularities lie over b 2 , while the points b 1 and b 3 are simple asymptotic values. Moreover, the critical point has multiplicity n 2 + n 3 − 1, and the equation f (λ) = b 2 has further n 1 + n 4 simple solutions. For fixed n = n 2 + n 3 + 1 and m = n 1 + n 4 + 1, there are m × n distinct line complexes and thus functions in F 1 . As for the line complex in Fig. 6 a similar analysis can be performed. One can read that the critical point lies over b 1 , two transcendental singularities lie over b 2 , and the other two over b 3 . Moreover, the critical point has multiplicity n 1 + n 2 + n 3 + n 4 + 1, the equation f (λ) = b 3 has n 1 + n 3 simple solutions, and the equation f (λ) = b 2 has n 2 + n 4 simple solutions. We now prove that a function belongs to the family F 1 if and only if f is the ratio of two solutions of a Hermite I or Hermite II harmonic oscillator. Theorem 3.3. The ratio of two particular solutions of a (m, n) Hermite I oscillator or to a (m, n) Hermite II oscillator belongs to the family F 1 with, in the case I, m = n 1 + n 4 + 1 and n = n 2 + n 3 + 1, and in the case II, m = n 2 + n 3 + 1 and n = n 1 + n 4 + 1.
Proof . The first part of the theorem follows by construction of the family F 1 . We now prove the second part. Since f belongs to F 1 , it has a unique critical point of multiplicity n 2 + n 3 . Around the critical point λ = 0, − 1 2 S(f ) = n 2 −1 4λ 2 + O( 1 λ ) where n = n 2 + n 3 . Moreover, f has four transcendental singularities and therefore − 1 2 S(f )(λ) is of the form P 2 (λ) + c −1 λ + n 2 −1 4λ 2 for some c −1 ∈ C and some quadratic polynomial of the form P 2 (λ) = λ 2 + c 1 λ + c 0 .
Denote S(f ) 1 : f → C the coefficient of the linear term of the Schwarzian derivative of f . The mapping is a bijection between the set F m,n and the set of the roots of generalised Hermite polynomial H m,n (z). has exactly m real roots when n is odd, and none when n is even.
Proof . About the order of H m,n (z). By the previous corollary, the number of roots of H m,n coincides with the number of distinct Stokes complexes such that m = n 1 + n 4 + 1 and n = n 2 + n 3 + 1; as it was already noted, this number equals m × n.
Concerning the number of real roots. By construction of the line complexes (Fig. 4), if f ∈ F m,n , has indices {n i }, then its conjugate, defined as f (λ), belongs to F m,n with indices {n i }, given by n 1 = n 1 , n 2 = n 3 , n 3 = n 2 and n 4 = n 4 . In particular f is self-conjugated, i.e., f is a real analytic function, if and only if n 2 = n 3 .
Clearly a is a real root iff (a, b) (b is the coefficient of the Laurent expansion at z = a defined in (1.5)) are reals iff the Schwarzian derivative of f is real iff f is real.
Therefore the number of real roots is equal to the number of real functions in F m,n . Since n = n 2 +n 3 +1, the constraint n 2 = n 3 has one and only one solution if n is odd, and no solutions if n is even. Therefore, if n is even, there are no normalised real functions f and thus H m,n (z) has no real roots.
On the other hand, if n is odd, the number of real functions equals the number of non-negative pairs of integers n 1 , n 4 such n 1 + n 4 = m − 1. This number is clearly m.
We now prove the analogue of Theorem 3.3 for Hermite III oscillators.
Theorem 3.6. The ratio of two particular solutions of a level (m, n) Hermite III oscillator belongs to the family F 2 with m = n 1 + n 3 + 1 and n = n 2 + n 4 + 1.
Proof . The proof is almost identical to the proof of Theorem 3.3 and therefore omitted.

Okamoto oscillators
In this subsection we discuss the Okamoto case. We identify a family of meromorphic functions to which the oscillators related to roots of the generalised Okamoto polynomials belong. Firstly, it is helpful to apply the following change of variables, This equation has a regular singular point at x = 0 with indices 1 2 ±3θ 0 , and an irregular singular point at x = ∞ of Poincaré rank 11.
Definition. For m, n ∈ Z, we call the anharmonic oscillator (3.5) (or its potential) with θ 0 = 1 2 n − 1 6 and θ ∞ = 1 2 (2m + n + 1), a level (m, n) Okamoto oscillator, if it Stokes multipliers satisfy Definition 3.7. We say that a meromorphic function f belongs to the class F n 3 , n ∈ Z if it has 13 singularities: 12 asymptotic values and 1 critical point. Moreover • The asymptotic values w 0 , . . . , w 11 in the Stokes sectors Ω 10 k , k = 0, . . . , 11 are Therefore, upon rescaling f → E 12 /E 22 f , it is easy to see that the Stokes data (3.6) imply that f takes the asymptotic value w k defined in (3.7), in the Stokes sector Ω k , for k ∈ Z 12 . The symmetry (3.8) is easily deduced from the fact that the potential satisfies V 10 (ζx) = ζV 10 (x).
Conversely, let f belong to F n 3 for some n ∈ Z. Then, by Theorem 3.2, it is clear that V (x) := − 1 2 S(f )(x) defines a Laurent polynomial, of degree 10. Furthermore, by the symmetry (3.8), we have V (ζx) = ζV (x), hence V (x) is a Laurent polynomial with only 1 (mod 3) degree terms.
The description of the correspondence in the above theorem is weaker than the ones in the Hermite cases, as the value of m is not understood on the coverings side of the correspondence. To be more concrete, let us discuss the special case n = 0, i.e., f has no critical points, for which it is easy to work out all possible line complexes for the F 0 3 family (up to rotation by π 2 i). Indeed in such case the line complex must be given by Fig. 7, for some p, q, r ∈ N, where we used b 1 = 1, b 2 = ζ, b 3 = ζ −1 and the Jordan curve γ equal to the unit circle. The question that remains, what values of p, q, r correspond to which value of m? We will not pursue this question further here.

Asymptotic analysis of Hermite I oscillators
This section of the article is dedicated to the asymptotic analysis of the zeros and poles of rational solutions of Hermite type, that is to say roots of the generalised Hermite polynomials H m,n (z), in the asymptotic regime n bounded and m → ∞.
Our analysis is based on the solution of the inverse problem characterising zeros of Hermite I solutions of P IV , see Theorem 2.2. For later convenience, we restate the inverse problem using the variables {α = E − 1 2 a, E = 2m + n} introduced in equation (1.11) above.
We tackle first Theorem 1.2, whose proof is divided in four steps: 1. In Section 4.1 we analyse the no-logarithm condition and by doing so we express the unknown β as an n−valued function of the unknown α.
2. Section 4.2 is devoted to the large E limit of the solution ψ + subdominant at λ = 0.
3. In Section 4.3 we compute, by means of the WKB method, the large E limit of the solution ψ 0 .
4. In Section 4.4, we match the asymptotic expansions of ψ + and ψ 0 to solve the inverse monodromy problem in the large E limit and prove Theorem 1.2.
Afterwards we show, in Section 4.5, all above steps can be appropriately modified in order to prove Theorem 1.3. Finally in Section 4.6 we comment on the distribution of zeros in the edge.

Asymptotic analysis of the no-logarithm condition
Equation (4.1) has a resonant singularity at λ = 0 with exponents 1±n 2 . For generic values of the parameters {α, β}, the Frobenius expansion of dominant solutions contain logarithmic terms. The singularity is apparent in those cases when these logarithmic terms are absent.
The no-logarithm condition imposes a polynomial constraint on the coefficients of equation (4.1) which allow us to express β as an n-valued function of the variable α. The n branches of this functions, which we denote by β j (α), j ∈ J n , are asymptotic to ji(1−α 2 ) 1 2 , in the large E limit. Indeed, we have the following proposition.
Proposition 4.1. Fix a simply connected compact domain D in the α-plane, not containing the points α = ±1. Then there exists an E 0 > 0, and n analytic functions β j (α) = β j (α, E) on D × [E 0 , ∞), j ∈ J n , such that the following statements hold true: 1. For every α ∈ D and E ≥ E 0 , there exist exactly n distinct values β such that the resonant singularity of equation (4.1) is apparent, given by β j (α), j ∈ J n .

2.
For j ∈ J n , the branch β j (α) has the following asymptotic expansion Proof . As a first simplification we apply a change of variables ν = E The resulting equation reads Clearly the singularity of the latter equation is apparent if and only if the singularity of equation (4.1) is apparent. Let us now consider the existence of a dominant solution, without logarithmic terms, of (4.2), Following the Frobenius method, the coefficients γ k satisfy the recursion where The recursion (4.3) is overdetermined at k = n. It has a solution, and thus infinitely many, if and only if −βγ n−1 − γ n−2 + c 1 E −1 γ n−3 + c 2 E −2 γ n−4 = 0.
Solving equation (4.3) recursively for all γ k with k ≤ n − 1, the no-logarithm condition becomes a single polynomial constraint of order n inβ, of the form where Q n (β) is a polynomial of order n, and the Q n,j are certain polynomial expressions inβ and (1 − α 2 ) − 1 2 . By definition, the equation Q n (β) = 0 is the no-logarithm condition for the differential equation (4.2) with E set equal to ∞, namely The latter equation is the Whittaker equation in a non-standard normal form, see [50,Section 13.14]. In fact, the general solution of this ODE is a linear combination of the Whittaker functions M κ,µ (z) and W κ,µ (z) with κ = − 1 2 βi, µ = 1 2 n and z = 2iν. From the formula [50, equation (13.14.18)], it follows that the above ODE has an apparent singularity if and only if β = ji, for some j ∈ J n . The thesis now easily follows.
As a consequence of the latter proposition, we have solved half of the inverse monodromy problem. Indeed, we can now limit our study to the following inverse problem depending on just one unknown α.
Here the function r j (α, E) is an asymptotically irrelevant contribution.

Whittaker asymptotics near the origin
In this subsection we find an approximation for the solution ψ + (λ) of equation (4.4) that is valid uniformly in E, in some E-dependent neighbourhood of the origin, slowly shrinking as E → +∞. According to our result, the solution ψ + takes the asymptotic form (4.5b) Here F n,j is defined in (1.12). This approximation is valid for λ belonging to a domain which scales as E γ− 1 2 , for some γ < 1 2 , on which the modulus of the function Σ is uniformly bounded. We therefore consider subsets of the λ-plane, which take the form for some fixed positive real numbers s, c, γ with γ < 1 2 . On such domains we have the following estimate for the solution subdominant at zero of equation (4.4).
Proposition 4.2. Upon fixing a suitable domain D and E 0 > 0 as in Proposition 4.1 and j ∈ J n , positive real numbers s, c, γ with γ < 1 2 and defining Λ + as in (4.6), let ψ + (λ) be the subdominant solution at zero of (4.4) normalised as follows Then there exists an E 1 ≥ E 0 such that for all E ≥ E 1 , the solution ψ + (λ) has the following properties: (i) ψ + (λ) is a holomorphic function of α in D; (ii) There exists a constant C -independent of α and E -such that for all λ ∈ Λ + , where Σ, ϕ and χ are as defined in equations (4.5) with γ = min{1 − 2γ, γ}.
Proof . Part (i) of the proposition is a standard application of the regular perturbation theory for linear ODEs. We divide the proof of part (ii) into several steps. Firstly, we apply the change Next we compare the solution u E (ν) = ν n+1 2 (1 + O(ν)) with u ∞ (ν), the subdominant solution of the same equation after E has been set equal to +∞, As was already mentioned in the proof of Proposition (4.1), the latter equation is the Whittaker equation in disguise; the general solution of this ODE is a linear combination of the Whittaker functions M κ,µ (z), W κ,µ (z) with κ = − 1 2 j, µ = 1 2 n and z = 2iν. In particular We now compare the solutions u E (ν) and u ∞ (ν).
Since | 1 0 |tR(t)|dt is of order E −1 , we have, for E big enough, that ||K|| n is smaller than one, and hence the resolvent series converges. Therefore, for E big enough, Let us call u(ν) the solution of (4.8) which solves the Cauchy problem u(1) = u E (1), u (1) = u E (1). Because of our discussion so far, for some bounded functionsc 1 ,c 2 . Note that u E also solves the integral equation To finish the proof of the claim we study the new integral operator on the Banach space H ρ of functions continuous on V ρ = V ∩ {|ν| ≤ ρ}, analytic in its interior, equipped with the standard supremum norm, which we denote by || · || ∞,ρ . From a standard WKB estimate, it follows that the general solution of the Whittaker equation (and its first derivatives) is, for large ν, asymptotic to a linear combination of the functions exp(±iν ∓ j 2 log ν). Therefore u(ν), u(t), K(ν, t) and its first derivatives are all uniformly bounded on V . The following estimate on the operator norm then easily follows, for some d 1 > 0, d 2 > 0, d 3 > 0 independent of α. The latter estimate together with the standard Volterra property leads to the following asymptotic estimate, for E big enough The same kind of estimate also holds for the difference of the derivatives because the kernel K has bounded derivatives. The claim now follows from these estimates and equation (4.11).
We have come to the third and last step of the proof, which is a straightforward application of the known asymptotic results of the Whittaker functions. From [50, equation (13.14.21)], we know there exists a C 4 such that in the domain V of the claim, the following estimates hold At the same time, if we let |ν| = E γ s, then, from the claim, we obtain that there exists a constant C 5 depending only on s such that the following estimates hold Combining the two sets of estimates, noticing that V ∩ {|ν| = E γ s} coincides with the set Λ + , after the change of variable λ = E − 1 2 (1 − α 2 ) − 1 2 ν, we obtain the thesis.
Observe that the best approximation is achieved with the choice γ = 1 3 , in which case the error scales with exponent − 1 3 .
Notice that no generality is lost in restricting to α < 1, as the roots of the generalised Hermite polynomials are symmetric under reflection in the imaginary axis.
In the proof of the above proposition, we assume that the reader has some familiarity with the WKB method. We begin the proof by analysing the turning points, i.e., zeros of the potential, of equation (4.4). A detailed dominant balance analysis, in the large E limit of the potential V A in equation (4.4), shows that asymptotically there are two turning points in the right half-plane. One of order E 1 2 and the other of order E − 1 2 , which we denote respectively by λ + and λ − . In fact we have In our analysis only λ + plays a role because the phase-shift originating from the turning point λ − is already taken into account in the Whittaker-like asymptotic, see Proposition 4.2.
The WKB theory is based around the analysis of the phase function where V A is the potential in equation (4.4). We compute the asymptotic behaviour of the phase function in the following lemma. (1 − α)}. Then the phase function S A admits the following asymptotic expansion in E, on H, for any ε > 0.
Moreover, f ix positive constants s and γ < 1 2 and let λ = E − 1 2 +γ s, then S A (λ) admits the expansion where γ = min{γ, 1 − 2γ}. In order to understand the WKB computation it is useful to draw the Stokes graph of the potential, that is the union of the Stokes lines λ λ * √ V A dµ = 0 where λ * is any turning point of V A (λ); we refer to [37] for more details.
We are interested only in the Stokes lines emanating from λ + , which can be read off formula (4.14), see Fig. 8.
Of the three Stokes lines emanating from λ + , one is the boundary between the Stokes sector Σ 0 and Σ +1 , another one is the boundary between the Stokes sector Σ 0 and Σ −1 , and the third one is the boundary between Σ +1 and Σ −1 . Notice that the region Λ + , where the behaviour of ψ + is known from Proposition 4.2, lies in the proximity of the third Stokes line, along which the solutions are oscillatory in nature 3 .
Therefore in order to compare ψ 0 and ψ + we need to compute the asymptotic behaviour of ψ 0 in the union of the Stokes domains Σ 1 and Σ −1 . To this aim we express ψ 0 as a linear combination of the solutions ψ ±1 (λ) subdominant in the Stokes domains Σ ±1 . Before doing so we need to define the error function ρ ±1 that controls the WKB estimates, Here : [0, 1) → C is any path such that: (i) (0) = λ; (ii) ± S A ( (t)) is monotonically increasing; (iii) lim t→1 S A ( (t)) = ±∞; see [37] for more details. We have the following lemma. 3 Notice that the notion of Stokes sectors Σ k that we use in WKB approximation does not coincide with the notion of Stokes sectors Ω k used in Theorem 2.2. However for large λ their boundaries are asymptotic. Lemma 4.5. There exist subdominant solutions ψ ±1 in Σ ±1 of equation (4.4), uniquely characterised by the asymptotic behaviour Furthermore the following hold true: (i) Both these subdominant solutions depend holomorphically on α in D.
which finishes the proof of the lemma.
The error functions ρ ±1 of the WKB approximation are estimated in the following lemma.
Lemma 4.6. Given the hypothesis in Proposition 4.3, there exist constants E 0 , C such that, for E ≥ E 0 , the following inequality holds for all (λ, α) ∈ Λ + × D.
Proof . The proof is in Appendix C.

Matching and asymptotics of poles
In this subsection we complete the proof of Theorem 1.2, by comparing the asymptotic form of the solutions ψ + (λ) and ψ 0 (λ) of equation (4.4). These were computed in Propositions 4.2 and 4.3, on the respective domains Λ + and Λ 0 ; these domains satisfy Λ + ⊂ Λ 0 . According to these results, on the domain Λ + the solutions ψ 0 and ψ + share the same trigonometric form but for a generally different phase, see equations (4.7) and (4.12). The phase-difference is the following function of α, , as defined in (1.12). It plays a major role in our analysis. In fact, we can express the Wronskian W [ψ 0 , ψ + ] = ψ 0 ψ + − ψ 0 ψ + in terms of Φ(α; E).

Proposition 4.7.
Here D is a simply connected compact subset of the vertical strip −1 < α < 1 and ψ + (λ) and ψ 0 (λ) are the solutions of (4.4) studied in Proposition 4.2 and 4.3. For E big enough we have the following decomposition of the Wronskian, where Φ(α; E) is defined in (4.16) and R(α, E) is an analytic function of α ∈ D such that Proof . R is analytic because ψ 0 , ψ + and sin(Φ(α; E)) are analytic in D. The estimate is a direct consequence of Proposition 4.2 and 4.3, specialised to γ = 1 3 .
According to the proposition above, the asymptotic distribution of the zeros of H m,n (α) with n bounded is determined by the zeros of the transcendental equation sin(Φ(α; E)) = 0. Equivalently for some k, integer if m is odd, half-integer if m is even, whose range will be determined below.
Notice that for any compact subset of the horizontal strip −1 < α < 1, equation (4.18) is just a vanishing perturbation of the equation As a first step we make the Ansatz 1 − α = κE − 2 3 and look for κ of order E 0 that solves the inverse monodromy problem for Hermite I rational solutions of P IV , equation (4.1). Upon inserting the Ansatz in equation (4.1), redefining the unknown β → E 1 3 β, and discarding lower order contributions, equation (4.1) reads ψ (λ) = V κ (λ; κ, β, E, n)ψ(λ), We look for (κ, β) such that no-logarithm and quantisation conditions hold.
No-logarithm in the edge. Reasoning as in Proposition 4.1, we can express the unknown β as an n-valued function of the unknown κ. To this aim, we apply the change of coordinates ν = E − 1 6 λ. The resulting equation reads where β j (κ), j ∈ J n is one of the n solutions of the nologarithm condition for the equation The above equation does not belong to the any standard class of linear special functions listed in [50] or [16]. In particular, the solution of the no-logarithm condition for the latter equation does not seem to have a simple factorisation. For the sake of the present discussion we assume that for κ > 0, the functions β j (κ) are all distinct as in the case of the Whittaker equation. Airy-like model near the origin. Applying again the change of variable ν = E − 1 6 λ, and reasoning as in Proposition 4.2, we deduce that the solution ψ + is asymptotic to the solution u + (ν), subdominant at λ = 0, of the equation (4.20) The latter equation is an Airy equation with a centrifugal term. The solution u + (ν) is thus for generic values of κ dominant at +∞. However, we say that κ is an eigenvalue of equation (4.20) if u + (ν) is subdominant at +∞.
Matching. The asymptotic analysis of the solution ψ 0 simplifies in this regime. In fact, in the large E limit, the turning points λ + and λ − coalesce with the Fuchsian singularity λ = 0. Therefore the standard approximation ψ 0 ∼ V − 1 4 A e −S A is valid in a region close to λ = 0, where the approximation of ψ + by u + is also valid. Finally, reasoning as in Proposition 4.7, we deduce that, asymptotically, the solutions ψ + and ψ 0 are proportional if and only if κ is one of the eigenvalue of (4.20). 4 Extremal asymptotic. We have therefore arrived at a (preliminary) description for the edge asymptotic for roots with extreme real part, without entering into the details of estimating the error of the approximation: Let (κ, β) be such that the Airy-like equation (4.19) satisfies the no-logarithm condition and the quantisation condition lim ν→+∞ u + (ν) = 0. If E is big enough, there exists a root of the generalised Hermite polynomial H m,n (α) with the asymptotic behaviour 1 − α = κE − 2 3 + o(E − 2 3 ).
Remark 4.9. In the case n = 1, the above description coincides with the well known edgeasymptotic for roots of the Hermite polynomial [16]. Indeed, in the case n = 1, the only solution of the no-logarithm condition is β = 0: the equation (4.19) reduces to the standard Airy equation with potential 2λ − 2κ and u + is the solution of the Airy equation that vanishes at λ = 0. Therefore the eigenvalues equal κ n = −2 − 1 3 ε n , n ≥ 1 where ε n is the n-th zero of the Airy function on the imaginary axis.

Concluding remarks
All singularities of Painlevé IV rationals can be expressed in terms of zeros of generalised Hermite H m,n and Okamoto polynomials Q m,n . We have characterised those zeros by means of the inverse monodromy problem for an anharmonic oscillator of degree two. We have shown that these oscillators in turn characterise three special classes of infinitely sheeted coverings of the sphere. In the case of generalised Hermite polynomials we have also explicitly computed the monodromy representation of such coverings by means of line complexes; as a by-product we obtained the number of real roots. Finally we have computed the asymptotic distribution in the bulk of zeros of generalised Hermite polynomials in the regime m 0 and n fixed; we have also presented some preliminary results about the edge asymptotics.
Many questions about rational solutions of Painlevé IV remain open and we plan to address some of them by means of the methods developed in the present paper. These include, in the case of generalised Hermite polynomials, a precise description of zeros in the edge -in particular the cross-over between the bulk and the edge behaviour described by the large κ limit of equation (4.20) -and the asymptotic distribution of zeros in the more general regime r := m/n ∈ (0, 1] fixed and both m, n → ∞. The latter is in principle a straightforward extension of our results, even though the error analysis presents some new analytical challenges. The real testing ground of our methods is the analysis of Okamoto polynomials. On the theoretical side, we have yet to find a combinatorial classification of line complexes of the functions in the class F n 3 discussed in Section 3. Such a classification would lead, along the lines of Corollary 3.5, to compute the number of real zeros. We also plan to tackle the asymptotic analysis of the distribution of zeros of generalised Okamoto polynomials, but at this stage it is uncertain whether our methods allows to deal with the greater complexity of these polynomials. Similarly the generalised Okamoto polynomials Q m,n , where m, n ∈ Z, are defined via the recursion Q m+1,n Q m−1,n = 9 2 Q m,n Q m,n − (Q m,n ) 2 + 2z 2 + 3(2m + n − 1) Q 2 m,n , Q m,n+1 Q m,n−1 = 9 2 Q m,n Q m,n + (Q m,n ) 2 + 2z 2 + 3(1 − m − 2n) Q 2 m,n , with Q 0,0 = Q 1,0 = Q 0,1 = 1 and Q 1,1 = √ 2z. The polynomials Q m,0 and Q m,1 with m ∈ Z, were originally introduced by Okamoto [49], to describe the subcases of (1.7) where θ 0 ∈ ± 1 6 , ± 1 3 . Proposition A.1. For m, n ∈ N, the polynomial H m,n has only simple roots and none common with H m+1,n , H m,n+1 and H m−1,n+1 . Furthermore H m,n 1 2 z is monic of degree mn with integer coefficients and we have the symmetry H m,n (iz) = i mn H n,m (z).

A Generalised Hermite and Okamoto polynomials
For m, n ∈ Z, the polynomial Q m,n has only simple roots and none common with Q m+1,n , Q m,n+1 and Q m−1,n+1 . Furthermore Q m,n 1 √ 2 z is monic of degree d m,n = m 2 + n 2 + mn − m − n with integer coefficients and we have the symmetries Q m,n (iz) = i dm,n Q n,m (z), Q m,n (z) = Q −m−n+1,m (z).
By definition ρ 1 ≤ |R(V A , µ)dµ| where the oriented integration contour is such that along it the real part of the phase function, S A , is not decreasing and tends to +∞ To this aim, we fix some 0 < r < 1 − α and define the following integration contour where λ r is the intercept with real part E 1 2 r of the Stokes line γ S defined by the equation S(µ) = S(λ). The second integral in taken along the anti-Stokes line γ A , defined by the equation S(µ) = S(λ r ) and the inequality S(µ) ≥ S(λ r ); see Fig. 8.
We first analyse the integral along the anti-Stokes line γ A . By our choice of the endpoint λ r , along γ A we have that |µ| ≥ CE 1 2 for some C > 0. We make the change of variable ν = E − 1 2 µ and expand the integrand around E = ∞ to get where V 0 (ν) = ν 2 + 2αν − (1 − α 2 ). Since E − 1 2 λ r has a well-defined limit different from zero and R(V 0 , ν) is integrable at ∞, we conclude that the integral along γ A is O(E −1 ).
We tackle now the contribution from the integration along the Stokes line. We split the integration contour in two: We fix r > 0 and write Similarly, for any fixed positive t, t , in the domain E 1 6 t ≤ | λ| ≤ E 1 2 t , |R(V A , E)| ≤ C |λ|E − 3 2 for E large enough (depending on r, r ). This means that the second integral is O(E −1 ).