Abstract
In this note we initiate the probabilistic study of the critical points of polynomials of large degree with a given distribution of roots. Namely, let f be a polynomial of degree n whose zeros are chosen IID from a probability measure μ on \(\mathbb{C}\). We conjecture that the zero set of f ′ always converges in distribution to μ as n → ∞. We prove this for measures with finite one-dimensional energy. When μ is uniform on the unit circle this condition fails. In this special case the zero set of f ′ converges in distribution to that of the IID Gaussian random power series, a well known determinantal point process.
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References
Abdul Aziz. On the zeros of a polynomial and its derivative. Bull. Austral. Math. Soc., 31(2):245–255, 1985.
J. Bak and D. Newman. Complex Analysis. undergraduate Texts in Mathematics. Springer-Verlag, Berlin, 1982.
Hubert E. Bray. On the Zeros of a Polynomial and of Its Derivative. Amer. J. Math., 53(4):864–872, 1931.
A. Bharucha-Reid and M. Sambandham. Random Polynomials. Academic Press, Orlando, FL, 1986.
Branko Ćurgus and Vania Mascioni. A contraction of the Lucas polygon. Proc. Amer. Math. Soc., 132(10):2973–2981 (electronic), 2004.
N. G. de Bruijn. On the zeros of a polynomial and of its derivative. Nederl. Akad. Wetensch., Proc., 49:1037–1044 = Indagationes Math. 8, 635–642 (1946), 1946.
N. G. de Bruijn and T. A. Springer. On the zeros of a polynomial and of its derivative. II. Nederl. Akad. Wetensch., Proc., 50:264–270 = Indagationes Math. 9, 458–464 (1947), 1947.
Dimitar K. Dimitrov. A refinement of the Gauss-Lucas theorem. Proc. Amer. Math. Soc., 126(7):2065–2070, 1998.
Janusz Dronka. On the zeros of a polynomial and its derivative. Zeszyty Nauk. Politech. Rzeszowskiej Mat. Fiz., (9):33–36, 1989.
R. Durrett. Probability: Theory and Examples. Duxbury Press, Belmont, CA, third edition, 2004.
Kenneth Falconer. Fractal geometry. John Wiley & Sons Inc., Hoboken, NJ, second edition, 2003. Mathematical foundations and applications.
A. W. Goodman, Q. I. Rahman, and J. S. Ratti. On the zeros of a polynomial and its derivative. Proc. Amer. Math. Soc., 21:273–274, 1969.
Andreas Hofinger. The metrics of Prokhorov and Ky Fan for assessing uncertainty in inverse problems. Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II, 215:107–125 (2007), 2006.
André Joyal. On the zeros of a polynomial and its derivative. J. Math. Anal. Appl., 26:315–317, 1969.
N.L. Komarova and I. Rivin. Harmonic mean, random polynomials and stochastic matrices. Advances in Applied Mathematics, 31(2):501–526, 2003.
K. Mahler. On the zeros of the derivative of a polynomial. Proc. Roy. Soc. Ser. A, 264:145–154, 1961.
S. M. Malamud. Inverse spectral problem for normal matrices and the Gauss-Lucas theorem. Trans. Amer. Math. Soc., 357(10):4043–4064 (electronic), 2005.
M. Marden. Geometry of Polynomials, volume 3 of Mathematical Surveys and Monographs. AMS, 1949.
M. Marden. Conjectures on the critical points of a polynomial. Amer. Math. Monthly, 90(4):267–276, 1983.
Piotr Pawlowski. On the zeros of a polynomial and its derivatives. Trans. Amer. Math. Soc., 350(11):4461–4472, 1998.
Y. Peres and B. Virag. Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process. Acta Math., 194:1–35, 2005.
Q. I. Rahman. On the zeros of a polynomial and its derivative. Pacific J. Math., 41:525–528, 1972.
Bl. Sendov. Hausdorff geometry of polynomials. East J. Approx., 7(2):123–178, 2001.
Bl. Sendov. New conjectures in the Hausdorff geometry of polynomials. East J. Approx., 16(2):179–192, 2010.
È. A. Storozhenko. On a problem of Mahler on the zeros of a polynomial and its derivative. Mat. Sb., 187(5):111–120, 1996.
Q. M. Tariq. On the zeros of a polynomial and its derivative. II. J. Univ. Kuwait Sci., 13(2):151–156, 1986.
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Pemantle, R., Rivin, I. (2013). The Distribution of Zeros of the Derivative of a Random Polynomial. In: Kotsireas, I., Zima, E. (eds) Advances in Combinatorics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30979-3_14
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DOI: https://doi.org/10.1007/978-3-642-30979-3_14
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