Hankel Determinant Approach to Generalized Vorob’ev–Yablonski Polynomials and Their Roots

Abstract

Generalized Vorob’ev–Yablonski polynomials have been introduced by Clarkson and Mansfield in their study of rational solutions of the second Painlevé hierarchy. We present new Hankel determinant identities for the squares of these special polynomials in terms of Schur polynomials. As an application of the identities, we analyze the roots of generalized Vorob’ev–Yablonski polynomials and provide a partial characterization for the boundary curves of the highly regular patterns observed numerically in Clarkson and Mansfield (Nonlinearity 16(3):R1–R26, 2003).

Appendices

Appendix 1: Proof of the Miura Relation (2.7)

Remark 5.1

We draw the reader’s attention to the notation used in this section:

$$\begin{aligned} {{\mathbf {t}}}= & {} (t_1,t_2,t_3,t_4,\dots )\ ,\ \ \ {{\mathbf {t}}}_o= (t_1,0,t_3,0,t_5,\dots )\ ,\ \ \ \widetilde{{{\mathbf {t}}}}_o= (t_1, 0,2^2 t_3, 0, 2^4 t_5,\dots ),\ \ \ \\&\,\,\,t_j\in {\mathbb {C}}. \end{aligned}$$

Let $k,\ell \in {\mathbb {Z}}_{\ge 0}$, and introduce

$$\begin{aligned} \mu _{k}({{\mathbf {t}}})= & {} \frac{1}{2\pi \mathrm {i}}\oint _{S} z^k \mathrm{e}^{\vartheta (z)}\frac{{\mathrm d}z}{z},\ \ \vartheta (z) = \vartheta (z;{{\mathbf {t}}}) = \sum _{j\ge 1} \frac{t_j}{z^j},\ \ \ \ \ \\ \Delta _{n,\ell }( {{\mathbf {t}}})= & {} \det \big [\mu _{\ell + j+k-2}({{\mathbf {t}}})\big ]_{j,k=1}^n,\ \ \Delta _{0,\ell }({{\mathbf {t}}})\equiv 1, \end{aligned}$$

where $S\subset {\mathbb {C}}$ denotes the unit circle traversed in counterclockwise orientation. Recalling (1.5), we see that

$$\begin{aligned} \mu _k({{\mathbf {t}}})=\frac{1}{k!}\frac{{\mathrm d}^k}{{\mathrm d}w^k}\exp \Bigg [\sum _{j\ge 1}t_jw^j\Bigg ]\Bigg |_{w=0}=h_k({{\mathbf {t}}}), \end{aligned}$$

and thus with (1.4),

$$\begin{aligned} \Delta _{n,\ell }({{\mathbf {t}}})=s_{(\ell +n+1)^n}({{\mathbf {t}}}). \end{aligned}$$

In particular, by Lemma 3.1, we know that for the special value $\ell =0$, we have the identity

$$\begin{aligned} \Delta _{n,0} ({{\mathbf {t}}}_o) =s_{(n+1)^n}({{\mathbf {t}}}_o)=2^{-n^2}s_{\delta _n}^2\big (\,\widetilde{{{\mathbf {t}}}}_o\big ). \end{aligned}$$

Next, let $\{p_{n,\ell }(z)\}_{n\ge 0}$ be the monic orthogonal polynomials associated with the measure

$$\begin{aligned} {\mathrm d}\nu _{\ell }(z)=\frac{1}{2\pi \mathrm {i}}z^\ell \mathrm{e}^{\vartheta (z)}\frac{{\mathrm d}z}{z},\ \ z\in S,\ \ell \in {\mathbb {Z}}_{\ge 0};\ \ \ \ \ \oint _Sp_{n,\ell }(z)p_{m,\ell }(z){\mathrm d}\nu _{\ell }(z)={\widehat{h}}_n\delta _{nm}. \end{aligned}$$

It is well known [11] that the matrix

$$\begin{aligned} \Gamma _{n,\ell }(z)= & {} \begin{bmatrix} p_{n,\ell }(z)&\frac{1}{2\pi \mathrm {i}}\oint _Sp_{n,\ell }(w)\frac{{\mathrm d}\nu _{\ell }(w)}{w-z}\\ \gamma _{n-1,\ell }\,p_{n-1,\ell }(z)&\frac{\gamma _{n-1,\ell }}{2\pi \mathrm {i}}\oint _Sp_{n-1,\ell }(w)\frac{{\mathrm d}\nu _{\ell }(w)}{w-z}\\ \end{bmatrix}, \ \ \nonumber \\&\,\,\,z\in {\mathbb {C}}\backslash S;\ \ \ \ \gamma _{n,\ell }=-2\pi \mathrm {i}\frac{\Delta _{n,\ell }({{\mathbf {t}}})}{\Delta _{n+1,\ell }({{\mathbf {t}}})} \end{aligned}$$

(5.1)

satisfies a Riemann–Hilbert problem; i.e. $\Gamma _{n,\ell }(z)$ is analytic for $z\in {\mathbb {C}}\backslash S$, and we have the conditions

$$\begin{aligned} \big (\Gamma _{n,\ell }(z)\big )_+= & {} \big (\Gamma _{n,\ell }(z)\big )_- \begin{bmatrix} 1&z^{\ell -1}\mathrm{e}^{\vartheta (z)}\\ 0&1 \end{bmatrix},\ \ z\in S;\ \ \ \ \nonumber \\ \Gamma _{n,\ell }(z)= & {} \left( I+\Gamma '_{n,\ell }(\infty )\frac{1}{z}+\mathcal {O}\left( z^{-2}\right) \right) z^{n\sigma _3},\ z\rightarrow \infty . \end{aligned}$$

Proposition 5.2

The following identities hold for the Hankel determinants: $\Delta _{n,\ell }({{\mathbf {t}}})$:

$$\begin{aligned} \frac{\Delta _{n,\ell +1}( {{\mathbf {t}}}-[z])}{\Delta _{n, \ell }({{\mathbf {t}}})} = (-1)^n\big (\Gamma _{n,\ell }(z)\big )_{11}, \ \qquad \ \ \frac{\Delta _{n,\ell -1}( {{\mathbf {t}}}+[z])}{\Delta _{n, \ell }({{\mathbf {t}}})} = (-1)^n\big (\Gamma _{n,\ell }(z)\big ) _{22}, \nonumber \\ \end{aligned}$$

(5.2)

and

$$\begin{aligned} \frac{\Delta _{n+1,\ell }({{\mathbf {t}}})}{\Delta _{n,\ell }({{\mathbf {t}}})} = -2\pi \mathrm {i}\,\big (\Gamma _{n,\ell }'(\infty )\big )_{12},\ \qquad \ \ \frac{\Delta _{n-1,\ell }({{\mathbf {t}}})}{\Delta _{n,\ell }({{\mathbf {t}}})} = \frac{\mathrm {i}}{2\pi }\,\big (\Gamma _{n,\ell }'(\infty )\big )_{21}, \end{aligned}$$

(5.3)

where [z] denotes the infinite vector of components $(z, \frac{z^2}{2}, \frac{z^3}{3}, \frac{z^4}{4},\ldots )$; i.e.

$$\begin{aligned} {{\mathbf {t}}}\mp [z]=\left( t_1\mp z,t_2\mp \frac{z^2}{2},t_3\mp \frac{z^3}{3},\ldots \right) . \end{aligned}$$

Proof

The two identities in (5.3) follow simply by inspection of the expression (5.1). As for the identities (5.2), the proof follows from Heine’s formula for the orthogonal polynomials and the observation that

$$\begin{aligned} w^{\ell } \exp \big [\vartheta (w; {{\mathbf {t}}}- [z])\big ]= & {} w^{\ell -1} (w-z) \exp \big [\vartheta (w; {{\mathbf {t}}})\big ],\ \ \ \ \\ w^{\ell } \exp \big [\vartheta (w; {{\mathbf {t}}}+ [z])\big ]= & {} \frac{w^{\ell +1}}{w-z} \exp \big [\vartheta (w; {{\mathbf {t}}})\big ]. \end{aligned}$$

Indeed, we have

$$\begin{aligned} \Delta _{n,\ell +1}({{\mathbf {t}}}-[z])&=\det \big [\mu _{\ell +j+k-1}({{\mathbf {t}}}-[z])\big ] _{j,k=1}^n\\&= \frac{1}{n!} \oint _{S^n} \prod _{ j<k} (w_j - w_k)^2 \prod _{j=1}^n w_j^{\ell } \exp \big [ \vartheta (w_j; {{\mathbf {t}}}-[z])\big ]\frac{{\mathrm d}w_j}{2\pi \mathrm {i}}\\&=\frac{1}{n!} \oint _{S^n} \prod _{j<k} (w_j - w_k)^2 \prod _{j=1}^n (w_j-z) {\mathrm d}\nu _{\ell }(w_j)\\&= (-1)^n\det \begin{bmatrix} \mu _{\ell }&\cdots&\mu _{\ell +n}\\ \vdots&\vdots \\ \mu _{\ell +n-1}&\mu _{\ell +2n-1}\\ 1&\cdots&z^n \end{bmatrix} = (-1)^np_{n,\ell }(z) \Delta _{n, \ell }({{\mathbf {t}}}),\qquad \quad \end{aligned}$$

where we used the well-known representation of orthogonal polynomials in terms of moment determinants (see, e.g., Proposition 3.8 in [11]). The second identity can be found in [4], but we can give here a direct derivation using Andreief’s identity [3]. Recall that ${\mathrm d}\nu _{\ell }(w)=w^{\ell {-}1}\mathrm{e}^{\vartheta (w)}{\mathrm d}w$. Then

$$\begin{aligned}&\oint _{S^n} \prod _{j<k}(w_k{-}w_j)^2 \prod _{j=1}^{n} \frac{{\mathrm d}\nu _{\ell }(w_j)}{w_j{-}z}\nonumber \\&\quad =\oint _{S^n} \det \Big [w_j^{k{-}1}\Big ]_{j,k=1}^n \det \Big [w_j^{k-1}\Big ]_{j,k=1}^n \prod _{j=1}^{n} \frac{{\mathrm d}\nu _{\ell }(w_j)}{w_j{-}z}\nonumber \\&\quad =\oint _{S^n} \det \Big [w_j^{k-1} \Big ]_{j,k=1}^n\det \bigg [\frac{w_j^{k-1}}{w_j-z}\bigg ]_{j,k=1}^n\prod _{j=1}^{n} {{\mathrm d}\nu _{\ell }(w_j)} . \end{aligned}$$

(5.4)

Multi-linearity allows us to replace the monic powers in the first determinant by the monic orthogonal polynomials $p_{j,\ell }(w)$, so that we obtain

$$\begin{aligned} (5.4)=\oint _{S^n} \det \Big [p_{k-1,\ell }(w_j)\Big ]_{j,k=1}^n\det \bigg [\frac{w_j^{k-1}}{w_j-z} \bigg ]_{j,k=1}^n\prod _{j=1}^{n} {{\mathrm d}\nu _{\ell }(w_j)}. \end{aligned}$$

(5.5)

Now, in the second determinant, we can subtract to the columns $2\le k\le n$ the multiple $z^{k-1}/(w_k-z)$ of the first column, thus obtaining

$$\begin{aligned} (5.5) = \oint _{S^n} \det \Big [p_{j-1,\ell }(w_k)\Big ]_{j,k=1}^n \det \begin{bmatrix} \frac{1}{w_1-z}&\frac{w_1-z}{w_1-z}&\cdots&\frac{w_1^{n-1}-z^{n-1}}{w_1-z}\\ \vdots&\vdots&\vdots \\ \frac{1}{w_n-z}&\frac{w_n-z}{w_n-z}&\cdots&\frac{w_n^{n-1}-z^{n-1}}{w_n-z} \end{bmatrix} \prod _{j=1}^n{\mathrm d}\nu _{\ell }(w_j).\nonumber \\ \end{aligned}$$

(5.6)

Now using Andreief’s identity, we obtain

but due to orthogonality, the matrix above has the following structure:

$$\begin{aligned} (5.6)= & {} n! \det \begin{bmatrix} \oint _Sp_{0,\ell }(w)\frac{{\mathrm d}\nu _{\ell }(w)}{w-z}&{\widehat{h}}_0&\star&\cdots&\star \\ \oint _Sp_{1,\ell }(w)\frac{{\mathrm d}\nu _{\ell }(w)}{w-z}&0&{\widehat{h}}_1&\star&\star \\ \vdots&\vdots&0&\ddots&\star \\ \vdots&\vdots&\vdots&\ddots&{\widehat{h}}_{n-2}\\ \oint _Sp_{n-1,\ell }(w)\frac{{\mathrm d}\nu _{\ell }(w)}{w-z}&0&0&\cdots&0 \end{bmatrix}\\= & {} (-1)^{n+1}n! \oint _S p_{n-1,\ell }(w)\frac{{\mathrm d}\nu _{\ell }(w) }{w-z}\prod _{j=0}^{n-2}{\widehat{h}}_j. \end{aligned}$$

However,

$$\begin{aligned} {\widehat{h}}_j = \oint _{S} p_{j,\ell }^2(w) {\mathrm d}\nu _{\ell }(w) = \frac{\Delta _{j+1,\ell }({{\mathbf {t}}})}{\Delta _{j, \ell }({{\mathbf {t}}})}, \end{aligned}$$

and therefore

$$\begin{aligned} \Delta _{n,\ell -2}({{\mathbf {t}}}+[z])= & {} (-1)^{n+1} \frac{\Delta _{n-1,\ell }({{\mathbf {t}}})}{\Delta _{0,\ell }({{\mathbf {t}}})}\oint _S p_{n-1,\ell }(w)\frac{{\mathrm d}\nu _{\ell }(w) }{w-z}\\ {}= & {} (-1)^n\Delta _{n,\ell }({{\mathbf {t}}})\big (\Gamma _{n,\ell }(z)\big )_{22}. \end{aligned}$$

$\square$

1.1 Dodgson–Hirota Bilinear Identity

Consider the following matrix-valued function:

$$\begin{aligned} H_{n,\ell }(z;{{\mathbf {t}}}, \mathbf{s}) = \Gamma _{n,\ell -1}(z;{{\mathbf {t}}}) \begin{bmatrix} \mathrm{e}^{\vartheta (z;{{\mathbf {t}}})-\vartheta (z;\mathbf{s})}&0\\ 0&z^2 \end{bmatrix} \Gamma _{n,\ell +1}^{-1}(z;\mathbf{s}),\ \ z\in {\mathbb {C}}\backslash (S\cup \{0\}). \end{aligned}$$

A direct inspection using the jumps of $\Gamma _{n,\ell }$ shows that this matrix has no jumps on the contour S; however, it has an essential singularity at $z=0$ due to the presence of the exponentials. We can thus compute the contour integral below in two ways. First, by evaluation as a residue at infinity:

$$\begin{aligned} \frac{1}{2\pi \mathrm {i}}\oint _{|z|=R}H_{n,\ell }(z;{{\mathbf {t}}}, \mathbf{s})\frac{{\mathrm d}z}{z} = \begin{bmatrix} 1-\big (\Gamma _{n,\ell -1}'(\infty ;{{\mathbf {t}}})\big )_{12} \big (\Gamma _{n,\ell +1}'(\infty ;\mathbf{s})\big )_{21}&\star \\ \star&\star \end{bmatrix}, \end{aligned}$$

(5.7)

where the $\star$ indicates expressions that are not relevant to the steps below. Secondly, we evaluate the left-hand side in (5.7) as a residue at $z=0$, but we are only interested in the (11)-entry,

$$\begin{aligned} \frac{1}{2\pi \mathrm {i}}\oint _{|z|=R}\big (H_{n,\ell }(z;{{\mathbf {t}}}, \mathbf{s})\big )_{11}\frac{{\mathrm d}z}{z}=\mathop {\mathrm {res}}\limits _{z=0}\frac{1}{z}\mathrm{e}^{ \vartheta (z;\underline{t}) - \vartheta (z;\underline{s}) }\big (\Gamma _{n,\ell -1}(z;{{\mathbf {t}}})\big )_{11}\big (\Gamma _{n,\ell +1}(z;\mathbf{s})\big )_{22}. \end{aligned}$$

Hence with (5.7) and Proposition 5.2,

$$\begin{aligned} \mathop {\mathrm {res}}\limits _{z=0}\frac{1}{z}\mathrm{e}^{ \vartheta (z;\underline{t}) - \vartheta (z;\underline{s}) }\frac{\Delta _{n,\ell }({{\mathbf {t}}}-[z])\Delta _{n,\ell }(\mathbf{s}+[z])}{\Delta _{n,\ell -1}({{\mathbf {t}}})\Delta _{n,\ell +1}(\mathbf{s})}= 1-\frac{\Delta _{n+1,\ell -1}({{\mathbf {t}}})\Delta _{n-1,\ell +1}(\mathbf{s})}{\Delta _{n,\ell -1}({{\mathbf {t}}})\Delta _{n,\ell +1}(\mathbf{s})}, \end{aligned}$$

or equivalently,

$$\begin{aligned}&\mathop {\mathrm {res}}\limits _{z=0}\frac{1}{z}\mathrm{e}^{ \vartheta (z;\underline{t}) - \vartheta (z;\underline{s}) }\Delta _{n,\ell }({{\mathbf {t}}}-[z])\Delta _{n,\ell }(\mathbf{s}+[z])\nonumber \\&\quad =\Delta _{n,\ell -1}({{\mathbf {t}}})\Delta _{n,\ell +1}(\mathbf{s})-\Delta _{n+1,\ell -1}({{\mathbf {t}}})\Delta _{n-1,\ell +1}(\mathbf{s}). \end{aligned}$$

(5.8)

Remark 5.3

Identity (5.8) closely resembles a “Hirota” version of the classical Dodgson determinantal identity, for if we set ${{\mathbf {t}}}=\mathbf{s}$, then (5.8) reduces to the Dodgson identity for Hankel determinants,

$$\begin{aligned} \Delta _{n,\ell }^2 = \Delta _{n,\ell -1}\, \Delta _{n,\ell +1}- {\Delta _{n+1,\ell -1} \Delta _{n-1,\ell +1}}. \end{aligned}$$

(5.9)

We now rewrite Eq. (5.8) with the substitution ${{\mathbf {t}}}\mapsto {{\mathbf {t}}}+ \mathbf{h},\mathbf{s} = {{\mathbf {t}}}-\mathbf{h}$ and define

$$\begin{aligned} HD_{n,\ell } ({{\mathbf {t}}},\mathbf{h})&= \mathop {\mathrm {res}}\limits _{z=0}\left( \frac{1}{z} \mathrm{e}^{2 \vartheta (z;\mathbf{h})} \Delta _{n,\ell }({{\mathbf {t}}}+\mathbf{h} -[z]) \Delta _{n, \ell }({{\mathbf {t}}}-\mathbf{h} + [z])\right) \\&- \Delta _{n,\ell -1}({{\mathbf {t}}}+\mathbf{h}) \Delta _{n,\ell +1}({{\mathbf {t}}}-\mathbf{h})- \Delta _{n+1,\ell -1}({{\mathbf {t}}}+\mathbf{h}) \Delta _{n-1,\ell +1}({{\mathbf {t}}}-\mathbf{h}) \end{aligned}$$

so that (5.8) can be written in the compact form

$$\begin{aligned} HD_{n,\ell } ({{\mathbf {t}}},\mathbf{h}) \equiv 0 \ ,\ \ \ \forall \, {{\mathbf {t}}}, \mathbf{h},\ \ \ \ \ \forall \, n,\ell \in {\mathbb {Z}}_{\ge 1}. \end{aligned}$$

(5.10)

For the rest of this section, we shall set all even times to zero; i.e., we choose ${{\mathbf {t}}}={{\mathbf {t}}}_o$. Now use Corollary 3.2 in conjunction with (5.9),

$$\begin{aligned} \Delta _{n,1}^2({{\mathbf {t}}}_o) = \Delta _{n,0}({{\mathbf {t}}}_o) \Delta _{n,2}({{\mathbf {t}}}_o) - \Delta _{n+1,0}({{\mathbf {t}}}_o) \Delta _{n-1,2}({{\mathbf {t}}}_o)=2(-1)^n\Delta _{n,0}({{\mathbf {t}}}_o)\Delta _{n+1,0}({{\mathbf {t}}}_o),\nonumber \\ \end{aligned}$$

(5.11)

and recall Lemma 3.1,

$$\begin{aligned} \Delta _{n,0}({{\mathbf {t}}}_o)=s_{(n+1)^n}({{\mathbf {t}}}_o)=2^{-n^2}s_{\delta _n}^2(\widetilde{{{\mathbf {t}}}}_o). \end{aligned}$$

Hence with (2.6) for $t_{2j+1}=0,j>N$ and $t_1=x$,

$$\begin{aligned} \Delta _{n,0}({{\mathbf {t}}}_o)=2^{-n^2}\mathrm{e}^{2g_n(x;\underline{t})},\ \ \ \ \ \ \ \Delta _{n+1,0}({{\mathbf {t}}}_o)=2^{-(n+1)^2}\mathrm{e}^{2W_n(x;\underline{t})+2g_n(x;\underline{t})}. \end{aligned}$$

(5.12)

Differentiating (5.10) with respect to $h_j$, we can derive a whole hierarchy of equations; however, we are only interested in one particular identity:

$$\begin{aligned} \frac{\partial ^2}{\partial h_1^2}HD_{n,\ell } ({{\mathbf {t}}}_o,\mathbf{h})\Big |_{\mathbf{h}=\mathbf{0}}= & {} - {\frac{\partial ^2 \Delta _{n+1,\ell -1} }{\partial t_{{1}}^2}} \Delta _{{n-1, \ell +1}} - \Delta _{{n+1, \ell -1}} {\frac{\partial ^2\Delta _{n-1,\ell +1} }{\partial t_{{1}}^2}} \nonumber \\&\,\,\,+\,2 {\frac{\partial \Delta _{{n+1, \ell -1}} }{\partial t_{{1}}}} {\frac{\partial \Delta _{{n-1, \ell +1}} }{\partial t_{{1}}}}\nonumber \\&\,\,\,+\, {\frac{\partial ^2 \Delta _{n,\ell -1} }{\partial t_{{1}}^2}} \Delta _{{n, \ell +1}} -2\, {\frac{\partial \Delta _{{n, \ell -1}} }{\partial t_{{1}}}} {\frac{\partial \Delta _{{n, \ell +1}} }{\partial t_{{1}}}} \nonumber \\&\,\,\,+ \, \Delta _{{n, \ell -1}} {\frac{\partial ^2 \Delta _{n,\ell +1} }{\partial t_{{1}}^2}} +2\, {\frac{\partial ^{2} \ln \Delta _{n,\ell } }{\partial {t_{{1}}}^{2}}} ( \Delta _{{n, \ell }})^2 =0, \quad \end{aligned}$$

(5.13)

and the argument of all determinants in the right-hand side equals ${{\mathbf {t}}}={{\mathbf {t}}}_o$. For $\ell =1$, with (5.13) and (3.5), (5.11), this leads to

$$\begin{aligned} 0= & {} \left( \Delta _{n+1,0} '' \Delta _{n,0} +\Delta _{n+1,0} \Delta _{n,0}'' - 2 \Delta _{n+1, 0} ' \Delta _{n,0}' \right) +\left( \ln \Delta _{n+1,0} + \ln \Delta _{n,0} \right) '' \Delta _{n+1,0} \Delta _{n,0},\ \ \\ (')= & {} \frac{\partial }{\partial t_1}, \end{aligned}$$

which can be rewritten as

$$\begin{aligned} 0= & {} \left( \frac{\Delta _{n+1,0} ''}{\Delta _{n+1,0}} +\frac{\Delta _{n,0}''}{\Delta _{n,0}} - 2\frac{ \Delta _{n+1, 0}'}{\Delta _{n+1,0}} \frac{\Delta _{n,0}' }{\Delta _{n,0}}\right) +\left( \ln \Delta _{n+1,0} + \ln \Delta _{n,0} \right) ''\\= & {} 2\frac{\partial ^2}{\partial t_1^2}\ln \big (\Delta _{n,0}\Delta _{n+1,0}\big ) +\left( \frac{\partial }{\partial t_1}\ln \frac{\Delta _{n,0}}{\Delta _{n+1,0}}\right) ^2, \end{aligned}$$

and after simplification with (5.12),

$$\begin{aligned} 2\,\partial _x^2g_n(x;\underline{t})=-\partial _x^2W_n(x;\underline{t})-\big (\partial _xW_n(x;\underline{t})\big )^2, \end{aligned}$$

which completes the proof of (2.7).

Appendix 2: The Outer Corners of the Regions $P_N$ for $N=2,3$

In this section, we offer a proof that the points (1.20) belong to the boundary of $P_N$. The proof is a verification that the inequalities for the effective potential are fulfilled at the particular values of a(x) determined in (4.8). These correspond in the a-plane to the points (1.20) in the x-plane. The proof is a simple deformation argument starting from large |a| (and hence also large x).

Observing various panes in Fig. 1 and using the ${{\mathbb {Z}}}_{2N+1}$ symmetry of the region, it is sufficient to show that the point

$$\begin{aligned} a_{0}^{[N]}= & {} \frac{1}{2} (-1)^N \left( 2N\left( \begin{array}{c}{2N}\\ {N}\end{array}\right) \right) ^{\frac{1}{2N+1}}\\ \Rightarrow x_{0}^{[N]}= & {} (-1)^N\left( (2N+1)\left( \frac{2N+1}{2N}\right) ^{2N}\left( \begin{array}{c}{2N }\\ {N}\end{array}\right) \right) ^{\frac{1}{2N+1}} \end{aligned}$$

(or rather its x–image) belongs to the boundary of $P_N$. This point is alternatively positive or negative, depending on the parity of N. Consider now in some detail the case $N=2$; then $a_{0}^{[2]} \simeq 0.944$ ($x_{0}^{[2]} \simeq 2.36021$). In this case, the polynomial P(z; a) (4.6) equals

$$\begin{aligned} P=P_2(z;a) = z^4 + \frac{1}{4} \frac{z^2}{a^3} - \frac{1}{2a}. \end{aligned}$$

Let $z_j^\pm (a),j=1, 2$ denote the roots of $P_2$. We know from the argument in Sect. 4.3 that for |a| large, the inequalities are fulfilled; as we deform a from larger absolute values to smaller ones, these inequalities can fail only if the sign of $\Phi (z_j^\pm (a); a)$ changes.

We now simply have to verify that the sign of $\Phi (z_j(a);a)$ remains constant as a decreases from $+\infty$ to the critical value $a_{0}^{[2]}$ (corresponding to x decreasing from $+\infty$ to the rightmost corner $x_{0}^{[2]}$). Since the four roots admit an explicit expression in terms of a, this verification is a simple exercise in calculus. To be more precise, one pair that we denote $z_2^\pm (a)$ is purely imaginary and lies on the zero-level set of $\Phi (z;a)$ identically for $a\in [a_{0}^{[2]}, \infty )$; this is not a cause of concern because it belongs to the level curve (in fact a straight line) joining $z=\pm \mathrm {i}a$ to $z=0$. The other pair $z_1^\pm (a)$ is real for $a\in [a_{0}^{[2]}, \infty )$. Then one can easily verify that

$$\begin{aligned} F(a)= \Phi \big (z_1^\pm (a);a\big ) \end{aligned}$$

is, depending on which of the two members of the pair, a monotone increasing/decreasing function of $a\in [a_{0}^{[2]}, \infty )$ and not changing sign. This verification uses Lemma 4.5 and the explicit expression for the roots, so that (for a real)

$$\begin{aligned} \frac{{\mathrm d}}{{\mathrm d}a} \Phi \big (z_j(a);a\big )= & {} - \mathfrak {R}\left( {\partial _x}\Phi (z;a) \frac{{\mathrm d}x}{{\mathrm d}a} \frac{\partial _a P(z;a)}{P'(z;a)} \Bigg |_{z=z_j(a)}\right) \nonumber \\= & {} \mathfrak {R}\left( \frac{\sqrt{z^2 + a^2}}{za} \frac{\partial _a P(z;a)}{P'(z;a)} \bigg |_{z=z_j(a)}\frac{{\mathrm d}x}{{\mathrm d}a}\right) . \end{aligned}$$

(6.1)

In Fig. 4, we display the graph of $\Phi (z_1(a);a)$ in the range $[a_{0}^{[2]},\infty )$; the monotonicity can be shown by inspecting the sign of (6.1); we leave the detail to the reader. The argument above can be repeated for $N=3$, but for larger N we were not able to find a unifying argument.

Fig. 4

The graph of the value of $\Phi$ at the saddle point $z_1$