Nuttall's Abelian integral on the Riemann surface of the cube root of a polynomial of degree 3

摘要

We study the field of orthogonal trajectories of a quadratic differential on the three-sheeted Riemann surface of the cube root of a polynomial of degree 3. These trajectories coincide globally with the level lines of the velocity potential of an incompressible fluid flowing to the surface through the infinitely remote point on one sheet and flowing out through the infinitely remote point on another. The statement of the problem is motivated by the task of finding the distribution of the poles of the Hermite-Pade approximants for two analytic functions with three common branch points, which is in its turn related to Nuttall's general conjecture.