Speaker
Description
We study the three-sheeted Riemann surface $S$ of genus $1$,corresponding to the algebraic function $w=\sqrt[3]{(z-a_1)(z-a_2)(z-a_3)}$ with pairwise distinct complex numbers $a_j$. There exists an abelian integral $G$ on $S$ which is regular on $S$, except for three points lying over infinity where the function Re $G$ has prescribed singularities of logarithmic type.
The harmonic function Re $G$ induces a partition of $S$ into three sheets; it is called the Nuttall decomposition. Such decomposition plays an important role in investigation of the asymptotics of the rational Hermite-Pade approximants. According to the Nuttall conjecture, the sheet gluing lines and their projections onto the Riemann sphere define the convergence domains for the Hermite-Pade approximants.
The differential-topological structure of the Nuttall decomposition depends essentially on the mutual location of the points $a_j$. A.I.Aptekarev and and D.N.Tulyakov (2016) described the structure for the case when the triangle with vertices at the points $a_j$ is sufficiently close to the regular one. We consider the general case and completely solve the problem for isosceles triangles with the apex angle less than $\pi/3$. In the talk we also discuss other cases.
The study was carried out at the expense of the grant of the Russian Science Foundation No.23-11-00066.
