On rational approximation of algebraic functions

Abstract

We construct a new scheme of approximation of any multivalued algebraic function $f(z)$ by a sequence $\{r_n(z){\}}_{n\in \mathbb{N}}$ of rational functions. The latter sequence is generated by a recurrence relation which is completely determined by the algebraic equation satisfied by $f(z)$. Compared to the usual Padé approximation our scheme has a number of advantages, such as simple computational procedures that allow us to prove natural analogs of the Padé Conjecture and Nuttall's Conjecture for the sequence $\{r_n(z){\}}_{n\in \mathbb{N}}$ in the complement $\mathbb{CP}^1\setminus \D_{f}$, where $\D_{f}$ is the union of a finite number of segments of real algebraic curves and finitely many isolated points. In particular, our construction makes it possible to control the behavior of spurious poles and to describe the asymptotic ratio distribution of the family $\{r_n(z){\}}_{n\in \mathbb{N}}$. As an application we settle the so-called 3-conjecture of Egecioglu {\em et al} dealing with a 4-term recursion related to a polynomial Riemann Hypothesis.

25 pages, 8 figures, LaTeX2e, revised version to appear in Advances in Mathematics

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Cited by in corpus (9)

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• Around a Conjecture of K. Tran
• Maximum Principle and Asymptotic Properties of Hermite--Padé Polynomials
• Special 5-term recurrence relations, Banded Toeplitz matrices, and Reality of Zeros
• On spectral asymptotic of quasi-exactly solvable quartic
• On Roots of Eigenpolynomials for Degenerate Exactly-Solvable Differential Operators
• On the use of rational-function fitting methods for the solution of 2D Laplace boundary-value problems
• On Asymptotics of Polynomial Eigenfunctions for Exactly-Solvable Differential Operators
• On the zeros of polynomials generated by rational functions with a hyperbolic polynomial type denominator