Location of the critical points of certain polynomials

Somjate Chaiya, Aimo Hinkkanen

Abstract


Let \(\mathbb{D}\) denote the unit disk \(\{z:|z|<1\}\) 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 \(P'\)  has exactly one critical point outside \(\overline{\mathbb{D}}\).

Keywords


Polynomial; critical point; anti-reciprocal.

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References


Boyd, D. W., Small Salem numbers, Duke Math. J. 44 (1977), 315–328.

Bertin, M. J., Decomps-Guilloux, A., Grandet-Hugot, M., Pathiaux-Delefosse, M., Schreiber, J. P., Pisot and Salem Numbers, Birkhauser Verlag, Basel, 1992.

Chaiya, S., Complex dynamics and Salem numbers, Ph.D. Thesis, University of Illinois at Urbana–Champaign, 2008.

Palka, Bruce P., An Introduction to Complex Function Theory, Springer-Verlag, New York, 1991.

Rahman, Q. I., Schmeisser, G., Analytic Theory of Polynomials, Clarendon Press, Oxford, 2002.

Salem, R., Power series with integral coefficients, Duke Math. J. 12 (1945), 153–173.

Salem, R., Algebraic Numbers and Fourier Analysis, D. C. Heath and Co., Boston, Mass., 1963.

Sheil-Small, T., Complex Polynomials, Cambridge University Press, Cambridge, 2002.

Walsh, J. L., Sur la position des racines des derivees d’un polynome, C. R. Acad. Sci. Paris 172 (1921), 662–664.




DOI: http://dx.doi.org/10.17951/a.2013.67.2.1-9
Data publikacji: 2015-07-15 00:00:00
Data złożenia artykułu: 2016-01-12 08:54:40

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