zbMATH — the first resource for mathematics

On the length of critical orbits of stable quadratic polynomials. (English) Zbl 1268.11155
Summary: We use the Weil bound of multiplicative character sums, together with some recent results of N. Boston and R. Jones, to show that the critical orbit of quadratic polynomials over a finite field of \(q\) elements is of length \(O(q^{3/4})\), improving upon the trivial bound \(q\).

11T06 Polynomials over finite fields
37P25 Dynamical systems over finite ground fields
11L40 Estimates on character sums
Full Text: DOI arXiv
[1] Nidal Ali, Stabilité des polynômes, Acta Arith. 119 (2005), no. 1, 53 – 63 (French). · Zbl 1088.11078 · doi:10.4064/aa119-1-4 · doi.org
[2] Mohamed Ayad and Donald L. McQuillan, Irreducibility of the iterates of a quadratic polynomial over a field, Acta Arith. 93 (2000), no. 1, 87 – 97. · Zbl 0945.11020
[3] Richard Crandall and Carl Pomerance, Prime numbers, 2nd ed., Springer, New York, 2005. A computational perspective. · Zbl 1088.11001
[4] Philippe Flajolet and Andrew M. Odlyzko, Random mapping statistics, Advances in cryptology — EUROCRYPT ’89 (Houthalen, 1989) Lecture Notes in Comput. Sci., vol. 434, Springer, Berlin, 1990, pp. 329 – 354. · Zbl 0747.05006 · doi:10.1007/3-540-46885-4_34 · doi.org
[5] D. Gomez and A. P. Nicolás, ‘An estimate on the number of stable quadratic polynomials’, preprint, 2010.
[6] Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. · Zbl 1059.11001
[7] Rafe Jones, Iterated Galois towers, their associated martingales, and the \?-adic Mandelbrot set, Compos. Math. 143 (2007), no. 5, 1108 – 1126. · Zbl 1166.11040 · doi:10.1112/S0010437X07002667 · doi.org
[8] Rafe Jones, The density of prime divisors in the arithmetic dynamics of quadratic polynomials, J. Lond. Math. Soc. (2) 78 (2008), no. 2, 523 – 544. · Zbl 1193.37144 · doi:10.1112/jlms/jdn034 · doi.org
[9] R. Jones and N. Boston, ‘Settled polynomials over finite fields,’ preprint, 2009. · Zbl 1243.11115
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.