Zero Distribution of Polynomials

  • Vladimir V. Andrievskii
  • Hans-Peter Blatt
Part of the Springer Monographs in Mathematics book series (SMM)


The classical theorems of Jentzsch and Szegö concern the limiting behavior of the zeros of the partial sums of a power series. More precisely, if
are the partial sums of a power series having finite positive radius of convergence ρ, then Jentzsch [91] proved that each point of the circle of convergence C ρ := {z: |z| = ρ} is a limit point of zeros of polynomials s n (z), n = 1, 2,... . Szegö [170] substantially improved this result by showing that there is a subsequence for which the zeros of the partial sums are uniformly distributed in angle; that is, if S(α, β) is the sector
, and Z n (A) denotes the number of zeros of s n in the set A, then
for all sectors S(α, β).


Maximum Principle Green Function Conformal Mapping Jordan Curve Type Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Historical Comments

  1. [40]
    A. Bloch, G. Pblya (1931): On the roots of certain algebraic questions. Proc. London Math. Soc., 33: 101–114.Google Scholar
  2. [157]
    E. Schmidt (1932): Ober algebraische Gleichungen vom Pôlya-Bloch-Typus, Sitzungsber. Preuss. Akad. Wiss. Phys.-Math. Kl., XXII: 321.Google Scholar
  3. [158]
    I. Schur (1933): Untersuchungen liber algebraische Gleichungen I. Bemerkungen zu einem Satz von E. Schmidt. Sitzungsber. Preuss. Akad. Wiss. Phys.-Math. Kl., X: 403–428.Google Scholar
  4. [172]
    G. Szegö (1934): Bemerkungen zu einem Satz von E. Schmidt über algebraische Gleichungen, Sitzungsber. Berl. Akad.: 86–98Google Scholar
  5. [67]
    T. Ganelius (1953): Sequences of analytic functions and their zeros. Ark. Mat. 3: 1–50.MathSciNetCrossRefGoogle Scholar
  6. [7]
    F. Amoroso, M. Mignotte (1996): On the distribution of the roots of polynomials. Ann. Inst. Fourier (Grenoble), 46: 1275–1291.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [76]
    R. Grothmann (1988): On the zeros of sequences of polynomials. J. Approx. Theory, 61: 351–359.MathSciNetCrossRefGoogle Scholar
  8. [38]
    H.-P. Blatt, E.B. Saff, M. Simkani (1988): Jentzsch-Szegd type theorems for the zeros of best approximants. J. London Math. Soc., 38: 307–316.MathSciNetCrossRefGoogle Scholar
  9. [77]
    R. Grothmann (1989): Ostrowski gaps, overconvergence and zeros of polynomials. Approximation Theory VI: Vol 1 (ed. C.K. Chui, L.L. Schumaker, and J.D. Ward ), Academic Press: 303–306.Google Scholar
  10. [78]
    R. Grothmann (1992): Interpolation Points and Zeros of Polynomials in Approximation Theory. Habilitationsschrift. Katholische Universität Eichstätt.Google Scholar
  11. [20]
    V.V. Andrievskii, H.-P. Blatt (1999): Erdfis-Turdn Type Theorems on Quasiconformal Curves and Arcs. J. Approx. Theory, 97: 334–365.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [31]
    H.-P. Blatt, R. Grothmann (1991): Erdös-Turcin theorems on a system of Jordan curves and arcs. Constr. Approx., 7: 19–47.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [151]
    P.C. Rosenbloom (1955): Distribution of zeros of polynomials. Lectures on Functions of a Complex Variable. Ann Arbor: 265–286.Google Scholar
  14. [164]
    P. Sjögren (1972): Estimates of mass distributions from their potentials and energies. Ark. Mat., 10: 59–77.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Vladimir V. Andrievskii
    • 1
  • Hans-Peter Blatt
    • 2
  1. 1.Department of Mathematics and Computer ScienceKent State UniversityKentUSA
  2. 2.Mathematisch-Geographische FakultätKatholische Universität EichstättEichstättGermany

Personalised recommendations