Geometry of Hermite-Padé approximants for system of functions $\{f,f^2\}$ with three branch points

In the problem on asymptotic of Hermite-Padé approximants for two analytic functions with branch points an algebraic function of the third order appears as the Cauchy transform of the limiting measure of poles distributions of the approximants. In general stuation this statement is known as the Nuttall’s conjecture. Our goal is, assuming that this conjecture holds true to describe the algebraic functions for the case when approximated two functions have common three branch points. In this preprint we discuss statement of the problem, general approaches to its solutions, and we carry out analysis of the appearing algebraic functions of genus zero. We plan to consider the cases corresponding to the algebraic functions of higher genus in the future paper.