×

Dynamical irreducibility of polynomials modulo primes. (English) Zbl 1489.11177

For a given a field \(K\) and a polynomial \(f \in K[X]\) consider the sequence of polynomials defined by \(f^{(0)}(X)=X\) and \(f^{(n)}(X)=f(f^{(n-1)}(X))\) for \(n=1,2,3,\dots\). A polynomial \(f \in K[X]\) is called stable (or dynamically irreducible) if its iterates \(f^{(n)}(X)\), \(n=1,2,3,\dots\), are irreducible over \(K\). For \(f \in {\mathbb Q}[X]\) and a prime number \(p\) let \(\overline{f}_p \in {\mathbb F}_p[X]\) be the reduction of \(f\) modulo \(p\). For a given dynamically irreducible polynomial \(f \in {\mathbb Q}[X]\) of degree \(d \geq 2\) it is not known whether the set of primes \(p\) for which \(\overline{f}_p\) is dynamically irreducible over \({\mathbb F}_p\) is a finite set. Let \(P_f(Q)\) be the set of primes in \([Q,2Q]\) for which \(\overline{f}_p\) is dynamically irreducible over \({\mathbb F}_p\). The authors show that if \(f \in {\mathbb Q}[X]\) is such that its derivative is of the form \(g(X)^2(aX + b)\), with \(g(X) \in {\mathbb Z}[X]\), \(a, b \in {\mathbb Z}\), \(a \ne 0\), and \(-b/a\) is not a pre-periodic point of \(f\), then one has \[P_f(Q) \leq \frac{(\log \log \log \log Q)^{2+o(1)}}{\log \log \log Q} \cdot \frac{Q}{\log Q} \] as \(Q \to \infty\). (In particular, all quadratic polynomials have their derivatives of the form as required.) They also show that under the assumption of GRH the bound is stronger \[P_f(Q) = O\Big(\frac{Q}{\log Q \log \log Q}\Big).\] The tools involve some effective results from Diophantine geometry, the square-sieve of Heath-Brown and character sums.

MSC:

11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11N36 Applications of sieve methods
37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps

Citations:

Zbl 1442.11150
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ali, N., Stabilité des polynômes, Acta Arith., 119, 53-63 (2005) · Zbl 1088.11078 · doi:10.4064/aa119-1-4
[2] Ayad, M., McQuillan, D. L.: Irreducibility of the iterates of a quadratic polynomial over a field. Acta Arith. 93, 87-97 (2000) [Corrigendum: Acta Arith., 99 (2001), 97] · Zbl 0945.11020
[3] Benedetto, R.; DeMarco, L.; Ingram, P.; Jones, R.; Manes, M.; Silverman, JH; Tucker, TJ, Current trends and open problems in arithmetic dynamics, Bull. Am. Math. Soc., 56, 611-685 (2019) · Zbl 1468.37001 · doi:10.1090/bull/1665
[4] Bérczes, A.; Evertse, J-H; Györy, K., Effective results for hyper- and superelliptic equations over number fields, Publ. Math. Debr., 82, 727-756 (2013) · Zbl 1274.11085
[5] Bombieri, E.; Gubler, W., Heights in Diophantine geometry (2006), Cambridge: Cambridge University Press, Cambridge · Zbl 1115.11034
[6] Cox, D., Primes of the form \(x^2+ny^2\): Fermat, class field theory, and complex multiplication (1997), New York: Wiley, New York · Zbl 0956.11500 · doi:10.1002/9781118032756
[7] Ferraguti, A., The set of stable primes for polynomial sequences with large Galois group, Proc. Am. Math. Soc., 146, 2773-2784 (2018) · Zbl 1442.11150 · doi:10.1090/proc/13958
[8] Gómez, D.; Nicolás, AP, An estimate on the number of stable quadratic polynomials, Finite Fields Appl., 16, 6, 401-405 (2010) · Zbl 1222.11143 · doi:10.1016/j.ffa.2010.06.005
[9] Gómez, D.; Nicolás, AP; Ostafe, A.; Sadornil, D., Stable polynomials over finite fields, Rev. Mat. Iberoam., 30, 523-535 (2014) · Zbl 1319.11089 · doi:10.4171/RMI/791
[10] Granville, A., ABC allows us to count squarefrees, Int. Math. Res. Not., 19, 991-1009 (1998) · Zbl 0924.11018 · doi:10.1155/S1073792898000592
[11] Heath-Brown, DR, The square sieve and consecutive squarefree numbers, Math. Ann., 266, 251-259 (1984) · Zbl 0514.10038 · doi:10.1007/BF01475576
[12] Heath-Brown, D.R., Micheli, G.: Irreducible polynomials over finite fields produced by composition of quadratics. Rev. Mat. Iberoam. (2020) (to appear) · Zbl 1498.11233
[13] Iwaniec, H.; Kowalski, E., Analytic number theory (2004), Providence, RI: American Mathematical Society, Providence, RI · Zbl 1059.11001
[14] Jones, R., The density of prime divisors in the arithmetic dynamics of quadratic polynomials, J. Lond. Math. Soc., 78, 523-544 (2008) · Zbl 1193.37144 · doi:10.1112/jlms/jdn034
[15] Jones, R., An iterative construction of irreducible polynomials reducible modulo every prime, J. Algebra, 369, 114-128 (2012) · Zbl 1302.11086 · doi:10.1016/j.jalgebra.2012.05.020
[16] Jones, R.; Boston, N., Settled polynomials over finite fields, Proc. Am. Math. Soc., 140, 1849-1863 (2012) · Zbl 1243.11115 · doi:10.1090/S0002-9939-2011-11054-2
[17] Jones, R.; Boston, N., Errata to “Settled polynomials over finite fields”, Proc. Am. Math. Soc., 148, 913-914 (2020) · Zbl 1434.11226 · doi:10.1090/proc/14761
[18] Konyagin, S.; Shparlinski, IE, Quadratic non-residues in short intervals, Proc. Am. Math. Soc., 143, 4261-4269 (2015) · Zbl 1378.11012 · doi:10.1090/S0002-9939-2015-12584-1
[19] Langevin, M., ‘Cas dégalité pour le théorème de Mason et applications de la conjecture (abc),’, C. R. Acad. Sci. Paris Sér. I Math., 317, 441-444 (1993) · Zbl 0788.11027
[20] Lidl, R., Niederreiter, H.: Introduction to finite fields and their applications. Cambridge University Press, Cambridge (1986) · Zbl 0629.12016
[21] Montgomery, HL; Vaughan, RC, Multiplicative number theory I: Classical theory (2006), Cambridge: Cambridge University Press, Cambridge · Zbl 1245.11002 · doi:10.1017/CBO9780511618314
[22] Silverman, JH, The arithmetic of dynamical systems (2007), New York: Springer, New York · Zbl 1130.37001 · doi:10.1007/978-0-387-69904-2
[23] Tenenbaum, G.: Introduction to analytic and probabilistic number theory, Grad. Studies Math., vol. 163. American Mathematical Society, Providence, RI (2015) · Zbl 1336.11001
[24] von zur Gathen, J., Gerhard, J.: Modern computer algebra. Cambridge University Press, Cambridge (1999) · Zbl 0936.11069
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.