Skip to main content
SpringerLink
Log in

Critical points of polynomials

  • Published:
Acta Mathematica Hungarica Aims and scope Submit manuscript

Abstract

This paper is devoted to the problem of where the critical points of a polynomial are relative to their zeros. Classical and new developments are surveyed along with illustrative examples. The paper finishes with a short proof of the sector theorem of Sendov and Sendov.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. N. G. de Bruijn and T. A. Springer, On the zeros of a polynomial and of its derivative, II, Nederl. Akad. Wetensch., 50 (1947), 264–270, Indag. Math., 9 (1947), 458–464.

  2. Casas-Alvero, E.: Siebeck curves and two refinements of the Gauss-Lucas theorem. Math. Scand. 111, 12–41 (2012)

    Article  MathSciNet  Google Scholar 

  3. Díaz-Barrero, J.L., Egozcue, J.J.: A generalization of the Gauss-Lucas theorem. Czechoslovak Math. J. 58(133), 481–486 (2008)

    Article  MathSciNet  Google Scholar 

  4. Dimitrov, D.: A refinement of the Gauss-Lucas theorem. Proc. Amer. Math. Soc. 126, 2065–2070 (1998)

    Article  MathSciNet  Google Scholar 

  5. Echols, W.H.: Note on the roots of the derivative of a polynomial. Amer. Math. Monthly 27, 299–300 (1920)

    Article  MathSciNet  Google Scholar 

  6. Erdős, P., Niven, I.: On the roots of a polynomial and its derivative. Bull. Amer. Math. Soc. 54, 184–190 (1948)

    Article  MathSciNet  Google Scholar 

  7. C. F. Gauss, Collected Works, Teubner (Leipzig 1900–1903), vol. 3, p. 112; vol. 8, p. 32; vol. 9, p. 187

  8. Genchev, T., Sendov, B.: A note on the theorem of Gauss for the distribution of the zeros of a polynomial on the complex plane. Fiz.-Math. Sp. 1, 169–171 (1958). (in Bulgarian)

    Google Scholar 

  9. Grace, J.H.: The zeros of a polynomial. Proc. Cambridge Philos. Soc. 11, 352–357 (1902)

    MATH  Google Scholar 

  10. Jensen, J.L.W.V.: Recherches sur la théorie dés équations. Acta Math. 36, 181–195 (1913)

    Article  MathSciNet  Google Scholar 

  11. Linfield, B.Z.: On certain polar curves with their application to the location of the roots of the derivatives of a rational function. Trans. Amer. Math. Soc. 27, 17–21 (1920)

    MathSciNet  MATH  Google Scholar 

  12. F. Lucas, Propriétés géométriques des fractions rationelles, C. R. Acad. Sci. Paris, 77 (1874), 431–433; 78 (1874), 140–144; 78 (1874), 180–183; 78 (1874), 271–274

  13. Malamud, S.M.: Inverse spectral problem for normal matrices and the Gauss-Lucas theorem. Trans. Amer. Math. Soc. 357, 4043–4064 (2005)

    Article  MathSciNet  Google Scholar 

  14. M. Marden, Geometry of Polynomials, 2nd ed., Math. Surveys 3, Amer. Math. Soc. (Providence, RI, 1966)

  15. Nikolov, N., Sendov, B.: A converse of the Gauss-Lucas theorem. Amer. Math. Monthly 121, 541–544 (2014)

    Article  MathSciNet  Google Scholar 

  16. Pereira, R.: Differentiators and the geometry of polynomials. J. Math. Anal. Appl. 285, 336–348 (2003)

    Article  MathSciNet  Google Scholar 

  17. Rahman, Q.I., Schmeisser, G.: Analytic Theory of Polynomials. Clarendon Press (2002)

    MATH  Google Scholar 

  18. M. Ravichandran, Principal submatrices, restricted invertibility and a quantitative Gauss–Lucas theorem, Int. Math. Res. Not., (2020), 4809–4832.

  19. Richards, T.: Level curve configurations and conformal equivalence of meromorphic functions. Comput. Methods Funct. Theory 15, 323–371 (2015)

    Article  MathSciNet  Google Scholar 

  20. T. J. Richards, On approximate Gauss–Lucas theorems. arXiv:1706.05410

  21. T. Richards, Some recent results on the geometry of complex polynomials: the Gauss–Lucas theorem, polynomial lemniscates, shape analysis, and conformal equivalence, Complex Anal. Synerg. (to appear), arXiv:1910.01159

  22. Richards, T.J., Steinerberger, S.: Leaky roots and stable Gauss-Lucas theorems. Complex Var. Elliptic Equ. 64, 1898–1904 (2019)

    Article  MathSciNet  Google Scholar 

  23. A. Rüdiger, Strengthening the Gauss–Lucas theorem for polynomials with zeros in the interior of the convex hull, arXiv:1405.0689

  24. Sheil-Small, T.: Complex Polynomials. Cambridge University Press (2002)

    Book  Google Scholar 

  25. Sendov, B., Sendov, H.: On the zeros and critical points of polynomials with nonnegative coefficients: a nonconvex analogue of the Gauss-Lucas theorem. Constr. Approx. 46, 305–317 (2017)

    Article  MathSciNet  Google Scholar 

  26. Siebeck, J.: Über eine neue analytische Behandlungweise der Brennpunkte. J. Reine Angew. Math. 64, 175–182 (1864)

    MathSciNet  Google Scholar 

  27. Totik, V.: The Gauss-Lucas theorem in an asymptotic sense. Bull. London Math. Soc. 48, 848–854 (2016)

    Article  MathSciNet  Google Scholar 

  28. Totik, V.: Distribution of critical points of polynomials. Trans. Amer. Math. Soc. 372, 2407–2428 (2019)

    Article  MathSciNet  Google Scholar 

  29. V. Totik, A quantitative Gauss–Lucas theorem (manuscript)

  30. Van Vleck, E.B.: On the location of roots of polynomials and entire functions. Bull. Amer. Math. Soc. 35, 643–683 (1929)

    Article  MathSciNet  Google Scholar 

  31. J. L. Walsh, On the location of the roots of the derivative of a polynomial, in: C. R. Congr. Internat. des Mathématiciens, Strasbourg, 1920, 339–342; Ann. of Math. (2), 22 (1920), 128–144

Download references

Author information

Authors and Affiliations

  1. MTA-SZTE Analysis and Stochastics Research Group, Bolyai Institute, University of Szeged, Aradi v. tere 1, Szeged, 6720, Hungary

    V. Totik

Authors
  1. V. Totik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. Totik.

Additional information

Dedicated to József Szabados

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Totik, V. Critical points of polynomials. Acta Math. Hungar. 164, 499–517 (2021). https://doi.org/10.1007/s10474-021-01133-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10474-021-01133-x

Key words and phrases

Mathematics Subject Classification

Search

Navigation