On rational approximation of algebraic functions

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dc.creatorBorcea, Julius
dc.creatorBögvad, Rikard
dc.creatorShapiro, Boris
dc.date2004-09-20
dc.date2005-06-17
dc.date.accessioned2026-07-07T06:30:32Z
dc.date.available2026-07-07T06:30:32Z
dc.descriptionWe 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.
dc.description25 pages, 8 figures, LaTeX2e, revised version to appear in Advances in Mathematics
dc.identifierhttps://arxiv.org/abs/math/0409353
dc.identifierhttp://arxiv.org/abs/math/0409353
dc.identifierAdvances in Mathematics vol. 204:2 (2006), 448-480.
dc.identifierdoi:10.1016/j.aim.2005.06.002
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/98365
dc.subjectClassical Analysis and ODEs
dc.subjectComplex Variables
dc.subjectPrimary 30E10; Secondary 41A20, 41A21, 41A25, 42C05, 82B05
dc.titleOn rational approximation of algebraic functions
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