Speaker
Description
We study the Nuttall decomposition of a three-sheeted torus $T$ which is a Riemann surface $S$ of the algebraic function $w=\sqrt[3]{(z-a_1)(z-a_2)(z-a_3)}$. This decomposition is induced by a harmonic function $U$ with two logarithmic singularities, it is called bipolar Green function. It is important to describe this function and investigate its properties. With the help of Weierstrass elliptic functions, we construct the universal covering of $T$ by the complex plane $\mathbb{C}$ and study the harmonic function $\widetilde{U}$ on $\mathbb{C}$ corresponding to $U$. The sets $\widetilde{U}(z)=0$, $\widetilde{U}(e^{2\pi i/3}z)=0$ and $\widetilde{U}(e^{-2\pi i/3}z)=0$ fully define decomposition of the complex plane into three parts which induces the Nuttall decomposition of $T$. According to the Nuttall conjecture, this decomposition defines the convergence domains for the Hermite--Padé approximants.
The study was supported by the grant of the Russian Science Foundation No. 23-11-00066.
