

Shafarevich Seminar
October 18, 2016 15:00, Moscow, Steklov Mathematical Institute, room 540 (Gubkina 8)






Rational approximation of the algebraic functions and functional analogues of the Diophantine approximations
A. I. Aptekarev^{} ^{} Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, Moscow

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Abstract:
Let f be a germ (the power series expansion) of an
algebraic function at infinity. We discuss the limiting properties
of the convergent of a functional continued fraction with polynomial
coefficients for f (alternative name is diagonal Pade approximant or
best local rational approximant). If we compare this functional
continued fraction for f with the usual continued fraction (with
integer coefficients) for a real number, then the degree of the
polynomial coefficient is analogous to the value (magnitude) of the
integer coefficient. In our joint work with M. Yattselev [1], we
derived strong (or BernshteinSzegö type) asymptotics for the
denominators of the convergent of the functional continued fraction
for analytic function with a finite number of branch points (which
are in a generic position in the complex plane). One of the
applications following from this result is a sharp estimate for a
functional analog of the ThueSiegelRoth theorem. The bounds on
the incomplete quotients for the functional continued fractions of
the algebraic functions follows from thus result as well.
References

A. I. Aptekarev, M. L. Yattselev, “Pade approximants for functions with branch points – strong asymptotics of NuttallStahl polynomials”, Acta Math., 215:2 (2015), 217–280, arXiv: 1109.0332v2

