On the distribution of zeros of the Hermite-Pade polynomials for three algebraic functions $1,f,f^2$ and the global topology of the Stokes lines for some differential equations of the third order

摘要

The paper presents some heuristic results about the distribution of zeros ofHermite-Pade polynomials of first kind for the case of three functions$1,f,f^2$, where $f$ has the form $f(z): = \prod\limits_ {j = 1 } ^3 (z-a_j) ^{\alpha_j} $, $\alpha_j \in \mathbb C\setminus \mathbb Z $, $ \sum \limits_ {j= 1 } ^ 3 \alpha_j = 0 $, $ f (\infty) = 1 $. Answers are given in termsrelated to the problem of extreme minimum capacity of a plane Nuttallcondenser, two plates of which intersect ("hooked" for each other) in a fivepoints: the branch points $ a_1, a_2, a_3 $ of function $ f $, at $ v_1 = v $,where $ v = v (a_1, a_2, a_3) $ the Chebotarev point corresponding to triplepoints $ a_1, a_2, a_3 $, and another "unknown" at $ v_2 \neq v_1, a_1, a_2,a_3 $ (see Fig1-Fig3). The connection between the distribution of zeros of Hermite-Pade polynomialsand global topology of the Stokes lines and the asymptotic behavior ofLiouville-Green solutions of a class of homogeneous linear differentialequations of the third order containing a large parameter is the free term isestablished. The basic idea of the new and still heuristic approach is to reduce at firstsome theoretical potential vector equilibrium problem to the scalar problemwith the external field, and then use the general method of Gonchar-Rakhmanovdeveloped in 1987 for solving the Varga problem "about $ 1/9 $". It is supposed that in the general case of an arbitrary algebraic function $f $ it is imposible to constract the Nuttall condenser for a set of threefunctions $ 1, f, f ^ 2 $ without the knowledge of the structure of the Stahlcompact for function $ f $. Namely, the Nuttall condenser is constructed usingGreen's function $ g_ {D} (z, \infty) $ for the Stahl domain $ D $ and "core"of Stahl compact, which consists of "effective" branch points of $ f $ andcorresponding Chebotarev points.