The string equation for polynomials
The author extends harmonic moments of a domain and the string equation for conformal maps to general polynomials. Let \(f(\zeta)=\sum_{j=0}^{\infty}a_j\zeta^{j+1}\), \(a_0>0\), be analytic in a neighborhood of the closure of the unit disk \(\mathbb D=\{\zeta\in\mathbb C:|\zeta|<1\}\) and \(f'\neq0\) on \(\partial\mathbb D\). If \(f\) is univalent on \(\mathbb D\), then the harmonic moments for \(\Omega=f(\mathbb D)\) are \[ M_k=\frac{1}{\pi}\int_{\Omega}z^kdxdy,\;\;k=0,1,\;\dots\;. \] The harmonic moments can be viewed as local coordinates in the space of univalent functions, that is \(f(\zeta)=f(\zeta;M_0,M_1,\dots)\). For any two functions \(f(\zeta;M_0,M_1,\dots)\) and \(g(\zeta;M_0,M_1,\dots)\) which are analytic in a neighborhood of \(\partial\mathbb D\), set \[ \{f,g\}=\zeta\frac{\partial f}{\partial\zeta}\frac{\partial g}{\partial M_0}-\zeta\frac{\partial g}{\partial\zeta}\frac{\partial f}{\partial M_0}. \] The string equation asserts that \(\{f,f^*\}=1\) with \(f^*(\zeta)=\overline{f(1/\overline{\zeta})}\). In the case of polynomials, the main result, Theorem 1, says that the string equation makes sense and holds whenever \(f'\) and \({f'}^*\) have no common zeros. Let \(g(\zeta)=\sum_{j=0}^nb_j\zeta^j\) and \(h(\zeta)=\sum_{k=0}^nc_k\zeta^{-k}\) and let \(\omega_1,\dots,\omega_n\) be the zeros of \(g\). Then the meromorphic resultant \(\mathcal R(g,h)\) between \(g\) and \(h\) is related to the polynomial resultant \(\mathcal R_{\text{pol}}(g(\zeta),\zeta^nh(\zeta))\) for polynomials \(g(\zeta)\) and \(\zeta^nh(\zeta)\) as \[ \mathcal R(g,h)=\frac{h(\omega_1)\cdot\cdot\cdot h(\omega_n)}{h(\infty)^n}=\frac{1}{b_0^nc_0^n}\mathcal R_{\text{pol}}(g(\zeta),\zeta^nh(\zeta)). \] Theorem 1. With \(f(\zeta)=\sum_{j=0}^na_j\zeta^{j+1}\), \(a_0>0\), the identity \[ \frac{\partial(\overline M_n,\dots,\overline M_1,M_0,M_1,\dots,M_n)}{\partial(\overline a_n,\dots,\overline a_1,a_0,a_1,\dots,a_n)}=2a_0^{n^2+3n+1}\mathcal R(f',{f'}^*) \] holds generally. It follows that the derivative \(\partial f/\partial M_0\) makes sense whenever \(\mathcal R(f',{f'}^*)\neq0\), and then also the string equation \(\{f,f^*\}=1\) holds. This result is an interplay between the Poisson bracket, the resultant and the Jacobi determinant for the moments and the coefficients of \(f\).
scientific article; zbMATH DE number 7000058
scientific article