Root-counting measures of jacobi polynomials and topological types and critical geodesics of related quadratic differentials

Boris Shapiro, Alexander Solynin

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

Two main topics of this paper are asymptotic distributions of zeros of Jacobi polynomials and topology of critical trajectories of related quadratic differentials. First, we will discuss recent developments and some new results concerning the limit of the root-counting measures of these polynomials. In particular, we will show that the support of the limit measure sits on the critical trajectories of a quadratic differential of the form (Formula Presented). Then we will give a complete classification, in terms of complex parameters a, b, and c, of possible topological types of critical geodesics for the quadratic differential of this type.

Original languageEnglish
Title of host publicationTrends in Mathematics
PublisherSpringer International Publishing
Pages369-438
Number of pages70
Edition9783319524696
DOIs
StatePublished - Jan 1 2017

Publication series

NameTrends in Mathematics
Number9783319524696
ISSN (Print)2297-0215
ISSN (Electronic)2297-024X

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Keywords

  • Asymptotic root-counting measure
  • Critical trajectories
  • Jacobi polynomials
  • Quadratic differentials

Cite this

Shapiro, B., & Solynin, A. (2017). Root-counting measures of jacobi polynomials and topological types and critical geodesics of related quadratic differentials. In Trends in Mathematics (9783319524696 ed., pp. 369-438). (Trends in Mathematics; No. 9783319524696). Springer International Publishing. https://doi.org/10.1007/978-3-319-52471-9_23