On Functional Graphs of Quadratic Polynomials

References

• [Bellah et al. 16] E. Bellah, D. Garton, E. Tannenbaum, and N. Walton. “A Probabilistic Heuristic for Counting Components of Functional Graphs of Polynomials Over Finite Fields,” Involve, 11 (2018), 169–179 (see http://arxiv.org/abs/1609.07667).
   Google Scholar
• [Benedetto et al. 2013] R. L. Benedetto, D. Ghioca, B. Hutz, P. Kurlberg, T. Scanlon and T. J. Tucker, “Periods of rational maps modulo primes”, Math. Ann. 355 (2013), 637–660.
   Web of Science ®Google Scholar
• [Bridy and Garton 16] A. Bridy and D. Garton. “Dynamically Distinguishing Polynomials,” Res. Math. Sci. 4 (2017) (see http://arxiv.org/abs/1609.09186).
   Web of Science ®Google Scholar
• [Burnette and Schmutz 17] C. Burnette and E. Schmutz. “Periods of Iterated Rational Functions Over a Finite Field,” Intern. J. Number Theory 13:3 (2017), 1301–1315.
   Google Scholar
• [Flajolet and Odlyzko 82] P. Flajolet, A. Odlyzko. “The Average Height of Binary Trees and Other Simple Trees,” J. Comput. Syst. Sci. 25:3 (1982), 171–213.
   Google Scholar
• [Flajolet and Odlyzko 90] P. Flajolet, A. Odlyzko. “Random Mapping Statistics,” Lecture Notes Comput. Sci. 434 (1990), 329–354.
   Google Scholar
• [Flynn and Garton 14] R. Flynn and D. Garton. “Graph Components and Dynamics Over Finite Fields,” Intern. J. Number Theory 10 (2014), 779–792.
   Web of Science ®Google Scholar
• [Gao and Rodrigues 03] S. Gao and V. M. Rodrigues. “Irreducibility of Polynomials Modulo p via Newton Polytopes,” J. Number Theory 101 (2003), 32–47.
   Web of Science ®Google Scholar
• [Gilbert et al. 01] C. L. Gilbert, J. D. Kolesar, C. A. Reiter, and J. D. Storey. “Function Digraphs of Quadratic Maps Modulo p,” Fibonacci Quart. 39 (2001), 32–49.
   Web of Science ®Google Scholar
• [Konyagin et al. 16] S. V. Konyagin, F. Luca, B. Mans, L. Mathieson, M. Sha, and I. E. Shparlinski. “Functional Graphs of Polynomials Over Finite Fields,” J. Combin. Theory, Ser. B 116 (2016), 87–122.
   Web of Science ®Google Scholar
• [Lorenzini 96] D. Lorenzini, An Invitation to Arithmetic Geometry, Amer. Providence: Math. Soc., 1996.
   Google Scholar
• [Martin and Panario 16] R. S. V. Martin and D. Panario. “On the Heuristic of Approximating Polynomials Over Finite Fields by Random Mappings,” Intern. J. Number Theory 12 (2016), 1987–2016.
   Web of Science ®Google Scholar
• [Morton 96] P. Morton. “On Certain Algebraic Curves Related to Polynomial Maps,” Compositio Math. 103 (1996), 319–350; Corrigendum, Compositio Math. 147 (2011), 332–334.
   Web of Science ®Google Scholar
• [Morton and Patel 94] P. Morton and P. Patel. “The Galois Theory of Periodic Points of Polynomial Maps,” Proc. London Math. Soc. 68 (1994), 225–263.
   Web of Science ®Google Scholar
• [Morton and Vivaldi 95] P. Morton and F. Vivaldi. “Bifurcations and Discriminants for Polynomial Maps,” Nonlinearity 8 (1995), 571–584.
   Web of Science ®Google Scholar
• [NTL 16] NTL: A Library for doing Number Theory, 2016, http://shoup.net/ntl/.
   Google Scholar
• [Ostafe and Sha 16] A. Ostafe and M. Sha. “Counting Dynamical Systems Over Finite Fields,” Contemp. Math. v.669, Amer. Math. Soc., 2016, 187–203.
   Google Scholar
• [Pari 16] The PARI Group, PARI/GP version 2.9.0, Univ. Bordeaux, 2016, http://pari.math.u-bordeaux.fr/.
   Google Scholar
• [Peinado et al. 01] A. Peinado, F. Montoya, J. Muñoz, and A. J. Yuste. “Maximal periods of x2 + c in Fq,” in S. Boztaş and I. E. Shparlinski (Eds.): Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Lecture Notes in Comput. Sci. 2227, Springer, Berlin, 2001, pp. 219–228.
   Google Scholar
• [Ruppert 86] W. M. Ruppert. “Reduzibilität ebener kurven,” J. Reine Angew. Math. 369 (1986), 167–191.
   Google Scholar
• [Silverman 07] J. H. Silverman, The arithmetic of dynamical systems, New York: Springer Verlag, 2007.
   Google Scholar
• [Vasiga and Shallit 04] T. Vasiga, and J. O. Shallit. “On the Iteration of Certain Quadratic Maps Over GF(p),” Discr. Math. 277 (2004), 219–240.
   Web of Science ®Google Scholar
• [Zannier 97] U. Zannier. “On the Reduction Modulo p of an Absolutely Irreducible Polynomial f(x, y),” Arch. Math. 68 (1997), 129–138.
   Web of Science ®Google Scholar