A. I. Aptekarev, M. Yattselev, “The Gonchar–Chudnovskies conjecture and a functional analogue of the Thue–Siegel–Roth theorem”, Tr. Mosk. Mat. Obs., 2022, Volume 83, Issue 2,Pages <nobr>297

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The Gonchar–Chudnovskies conjecture and a functional analogue of the Thue–Siegel–Roth theorem

A. I. Aptekarev^a, M. Yattselev^b

^a Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, Moscow
^b Indiana University–Purdue University Indianapolis

Abstract: This article examines the Gonchar–Chudnovskies conjecture about the limited size of blocks of diagonal Padé approximants of algebraic functions. The statement of this conjecture is a functional analogue of the famous Thue–Siegel–Roth theorem. For algebraic functions with branch points in general position, we will show the validity of this conjecture as a consequence of recent results on the uniform convergence of the continued fraction for an analytic function with branch points. We will also discuss related problems on estimating the number of “spurious” (“wandering”) poles for rational approximations (Stahl’s conjecture), and on the appearance and disappearance of defects (Froissart doublets).

UDC: 517.538.5

MSC: 41A21, 30B40

Received: 05.07.2022