Row-Vortex Polynomials in Integrable Systems

• Row-vortex polynomials are a sequence of special monic polynomials arising in integrable systems and vortex equilibria, with roots exhibiting pronounced cyclic symmetry.
• They satisfy nontrivial bilinear and log-derivative recurrence relations that allow explicit recursive construction from the underlying generalized K₂–hierarchy.
• Their rich algebraic structure not only encodes equilibrium configurations in fluid dynamics but also bridges deep connections between algebra, analysis, and physical phenomena.

The row-vortex polynomials represent a distinguished sequence of special monic polynomials intimately connected to integrable hierarchies (notably, the generalized K₂–hierarchy), rational solutions of nonlinear partial differential equations such as the Sawada–Kotera and Kaup–Kupershmidt equations, and explicit configurations of point vortices with prescribed circulations in the plane. These polynomials, denoted $P_n(z)$ and accompanied by a companion sequence %%%%1%%%%, encode equilibrium configurations of point vortices with circulations $\Gamma$ and $-2\Gamma$. The roots of these polynomials, their recurrence and differential structures, and their symmetry and scaling properties display a rich interplay between algebraic, analytic, and physical aspects within the theory of integrable systems and fluid dynamics (Demina et al., 2011).

1. Definition and Normalization

The row-vortex polynomial sequence $\{P_n(z)\}$ is defined by the following normalization:

• $P_0(z) = 1$
• $P_1(z) = z$
• For $n \geq 2$, $P_n(z)$ is monic of degree

$$d_n = \deg P_n(z) = \frac{6n(n+1) - 1 + (-1)^n(2n+1)}{8}$$

• Each $P_n(z) = z^{d_n} +$ (lower-degree terms)

The companion sequence $\{Q_n(z)\}$ is likewise monic, with $Q_0(z) = 1$, $Q_1(z) = z$, and for $n \geq 2$,

$$\deg Q_n(z) = \frac{6n(n+1) + 1 + (-1)^{n+1}(2n+1)}{16}$$

These sequences classify polynomial representatives for stationary vortex configurations with circulations $\Gamma$ (from roots of $P_n$) and $-2\Gamma$ (from roots of $Q_n$) (Demina et al., 2011).

2. Recurrence and Differential–Difference Relations

The row-vortex polynomials satisfy nontrivial coupled bilinear and first-order difference relations.

Bilinear (generalized Hirota) relation ($\mu=2$):

$D_z^2[2]P \cdot Q := P'' Q - 4P' Q' + 4P Q'' = 0$

where $D_z^m[\mu]F \cdot G$ denotes the generalized Hirota derivative.

For neighboring indices:

• $$P_{n\pm 1}'' Q_n - 4P_{n\pm 1}' Q_n' + 4P_{n\pm 1} Q_n'' = 0$$
• $$P_{n}'' Q_{n\pm 1} - 4P_{n}' Q_{n\pm 1}' + 4P_{n} Q_{n\pm 1}'' = 0$$

Log-derivative recurrence:

\begin{align*} P_{n+1}' P_{n-1} - P_{n+1} P_{n-1}' &= \gamma_{n+1} Q_n^4 \ Q_{n+1}' Q_{n-1} - Q_{n+1} Q_{n-1}' &= \delta_{n+1} P_n^2 \end{align*} with explicit coefficients:

$$\gamma_{n+1} = 6n + 3 + (-1)^{n+1}2,\quad \delta_{n+1} = 6n + 3 + (-1)^n 4$$

These relations enable explicit recursive construction of $P_n$ and $Q_n$ for all $n \geq 0$. They encode the algebraic structure imposed by the underlying integrable hierarchy (Demina et al., 2011).

3. Ordinary Differential Equations and Hierarchical Structure

Each row-vortex polynomial $P_n(z)$ satisfies a linear ODE in $z$ whose order is determined by the member $N$ of the generalized $K_2$–hierarchy to which it is associated. For $N=1$, the polynomial satisfies:

$$\alpha_1\left(h_{zzzzz} + 15 h_z h_{zzz} + \tfrac{45}{4} h_{zz}^2 + 15 h_z^3\right) - z h_z - h = 0$$

where $h(z) = \frac{d}{dz}\ln P_n(z)$ and $\alpha_1$ is a parameter. For $N>1$, analogous higher-order equations arise, always tied to the algebraic structure of the corresponding integrable PDE (Demina et al., 2011).

A key feature is that matching the local expansion of $h(z)$ at infinity against asymptotics of the ODE leads to algebraic constraints on the power sums of the roots, dictating their distribution and symmetry properties.

4. Vortex Equilibria Interpretation

Row-vortex polynomials possess a direct physical interpretation: their roots yield explicit stationary equilibrium configurations for point vortices in the complex plane. Let $l_1$ vortices of circulation $\Gamma$ be located at $\{a_i\}$ and $l_2$ vortices of circulation $-2\Gamma$ at $\{b_j\}$. The generating polynomials

$$P(z) = \prod_{i=1}^{l_1} (z - a_i),\quad Q(z) = \prod_{j=1}^{l_2} (z - b_j)$$

satisfy the bilinear relation $P'' Q - 4P' Q' + 4P Q'' = 0$ precisely when the system is in stationary equilibrium ($\dot z_k^* = 0$). Thus, for coprime, monic solutions $(P,Q)$ of the relation, vortex locations $\{a_i\}$ and $\{b_j\}$ give an admissible physical configuration. The normalizations may be taken as $P = P_n$, $Q = Q_{n+1}$ or vice versa (Demina et al., 2011).

Explicit low-degree cases, e.g., $P_2(z) = z^5 + 36$ (roots forming a regular pentagon) and $P_4(z)$ (roots forming a regular 15-gon and one simple root at the origin), exemplify this interpretation.

5. Algebraic Structure of Roots

The roots of row-vortex polynomials display strong algebraic constraints. Writing

$$h(z) = \frac{d}{dz}\ln P_n(z) = \sum_{i=1}^{d_n} \frac{1}{z - a_i}$$

and expanding at infinity, the resulting power sums satisfy

• $\sum_i a_i^k = 0$ for $k = 1,2,3,4$
• $\sum_i a_i^5 = 3\alpha_1 d_n(d_n - 1)(d_n - 8)$ and, for the roots of $Q_n(z)$ (denoted $b_j$),
• $$\sum_j b_j^5 = 12 \alpha_1 d_n'(d_n' - 1)(d_n' - 2)$$ where $d_n$ and $d_n'$ are the degrees of $P_n$ and $Q_n$, respectively.

Only power sums with degree divisible by $5$ are non-vanishing, implying a pronounced cyclic (dihedral) symmetry in the root distribution. For representative values of $\alpha_1$, roots are found on regular star polygons. Exceptional high-multiplicity roots at the origin correspond to “coalesced” vortices (Demina et al., 2011).

6. Structural Properties and Symmetries

Several additional features characterize the row-vortex polynomials:

• Parity and $z \mapsto -z$ symmetry: Many polynomials exhibit dihedral symmetry in their root geometry.
• Scaling: Under $z \to \lambda z$ and $\alpha_1 \to \lambda^5 \alpha_1$, $$P_n(\lambda z; \lambda^5 \alpha_1) = \lambda^{d_n} P_n(z; \alpha_1)$$, indicating homogeneous scaling.
• Exceptional roots at $z=0$: Multiplicity can only be $1$ or $8$ (for $P_n$), $0$, $1$, or $2$ (for $Q_n$).
• Interlacing of zeros: For generic parameters, roots of $P_n$ and $Q_{n+1}$ are simple and lie on interlacing curves in $\arg z$.

These properties reflect the deep relationship between polynomial invariants, integrable PDE rational solutions, and classical vortex dynamics (Demina et al., 2011).

7. Context within Vortex Polynomials and Nonuniqueness

The row-vortex polynomials are situated in the broader landscape of “vortex polynomials” arising from equilibrium problems of point vortices. While the Adler–Moser polynomials have long been considered canonical solutions to the Tkachenko equation governing vortex equilibrium with arbitrary circulations, it has been established that multiple distinct polynomial families, including the row-vortex polynomials, also provide valid solutions. Notably, new polynomial solutions differing in degrees and root patterns from the Adler–Moser sequence arise through exponentiation and combination of “$\mu$-equation” pairs, such as $(P_n,Q_{n\pm1})$ (Demina et al., 2011).

A plausible implication is that the theory of vortex polynomials is significantly deeper and less rigid than previously assumed, with the row-vortex case illustrating new algebraic and geometric structures emergent from integrable hierarchies and their links to vortex configurations (Demina et al., 2011, Demina et al., 2011).