Location of the critical points of certain polynomials

Somjate Chaiya; Aimo Hinkkanen

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica (2013)

  • Volume: 67, Issue: 2
  • ISSN: 0365-1029

Abstract

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Let 𝔻 denote the unit disk { z : | z | < 1 } in the complex plane ℂ . In this paper, we study a family of polynomials P with only one zero lying outside 𝔻 ¯ .  We establish  criteria for P to satisfy implying that each of P and P '   has exactly one critical point outside 𝔻 ¯ .

How to cite

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Somjate Chaiya, and Aimo Hinkkanen. "Location of the critical points of certain polynomials." Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 67.2 (2013): null. <http://eudml.org/doc/289781>.

@article{SomjateChaiya2013,
abstract = {Let $\mathbb \{D\}$ denote the unit disk $\lbrace z:|z|<1\rbrace $ in the complex plane $\mathbb \{C\}$. In this paper, we study a family of polynomials $P$ with only one zero lying outside $\overline\{\mathbb \{D\}\}$.  We establish  criteria for $P$ to satisfy implying that each of $P$ and $P^\{\prime \}$  has exactly one critical point outside $\overline\{\mathbb \{D\}\}$.},
author = {Somjate Chaiya, Aimo Hinkkanen},
journal = {Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
keywords = {Polynomial; critical point; anti-reciprocal.},
language = {eng},
number = {2},
pages = {null},
title = {Location of the critical points of certain polynomials},
url = {http://eudml.org/doc/289781},
volume = {67},
year = {2013},
}

TY - JOUR
AU - Somjate Chaiya
AU - Aimo Hinkkanen
TI - Location of the critical points of certain polynomials
JO - Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2013
VL - 67
IS - 2
SP - null
AB - Let $\mathbb {D}$ denote the unit disk $\lbrace z:|z|<1\rbrace $ in the complex plane $\mathbb {C}$. In this paper, we study a family of polynomials $P$ with only one zero lying outside $\overline{\mathbb {D}}$.  We establish  criteria for $P$ to satisfy implying that each of $P$ and $P^{\prime }$  has exactly one critical point outside $\overline{\mathbb {D}}$.
LA - eng
KW - Polynomial; critical point; anti-reciprocal.
UR - http://eudml.org/doc/289781
ER -

References

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  1. Boyd, D. W., Small Salem numbers, Duke Math. J. 44 (1977), 315–328. 
  2. Bertin, M. J., Decomps-Guilloux, A., Grandet-Hugot, M., Pathiaux-Delefosse, M., Schreiber, J. P., Pisot and Salem Numbers, Birkhauser Verlag, Basel, 1992. 
  3. Chaiya, S., Complex dynamics and Salem numbers, Ph.D. Thesis, University of Illinois at Urbana–Champaign, 2008. 
  4. Palka, Bruce P., An Introduction to Complex Function Theory, Springer-Verlag, New York, 1991. 
  5. Rahman, Q. I., Schmeisser, G., Analytic Theory of Polynomials, Clarendon Press, Oxford, 2002. 
  6. Salem, R., Power series with integral coefficients, Duke Math. J. 12 (1945), 153–173. 
  7. Salem, R., Algebraic Numbers and Fourier Analysis, D. C. Heath and Co., Boston, Mass., 1963. 
  8. Sheil-Small, T., Complex Polynomials, Cambridge University Press, Cambridge, 2002. 
  9. Walsh, J. L., Sur la position des racines des derivees d’un polynome, C. R. Acad. Sci. Paris 172 (1921), 662–664. 

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