Decomposable polynomials in second order linear recurrence sequences

  • Clemens FuchsEmail author
  • Christina Karolus
  • Dijana KresoEmail author
Open Access


We study elements of second order linear recurrence sequences \((G_n)_{n= 0}^{\infty }\) of polynomials in \({{\mathbb {C}}}[x]\) which are decomposable, i.e. representable as \(G_n=g\circ h\) for some \(g, h\in {{\mathbb {C}}}[x]\) satisfying \(\deg g,\deg h>1\). Under certain assumptions, and provided that h is not of particular type, we show that \(\deg g\) may be bounded by a constant independent of n, depending only on the sequence.

Mathematics Subject Classification

11B37 11R09 12E99 39B12 



Open access funding provided by Austrian Science Fund (FWF). The work on this manuscript was supported by FWF (Austrian Science Fund) Grant No. P24574 and No. J3955.


  1. 1.
    Beals, R.M., Wetherell, J.L., Zieve, M.E.: Polynomials with a common composite. Isr. J. Math. 174, 93–117 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Beardon, A.F., Ng, T.W.: On Ritt’s factorization of polynomials. J. London. Math. Soc. 62, 127–138 (2000)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bilu, Yu.F.: Quadratic factors of \(f(x)-g(y)\). Acta Arith. 90, 341–355 (1999)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bilu, Yu.F., Tichy, R.F.: The Diophantine equation \(f(x) = g(y)\). Acta Arith. 95, 261–288 (2000)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Brownawell, W.D., Masser, D.W.: Vanishing sums in function fields. Math. Proc. Cambridge Philos. Soc. 100(3), 427–434 (1986)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dujella, A., Tichy, R.F.: Diophantine equations for second-order recursive sequences of polynomials. Q. J. Math. 52, 161–169 (2001)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fried, M.D.: On a conjecture of Schur. Michigan Math. J. 17, 41–55 (1970)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Fuchs, C.: On the Diophantine equation \(G_n(x)=G_m(P(x))\) for third order linear recurring sequences. Port. Math. (N.S.) 61, 1–24 (2004)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Fuchs, C., Pethő, A.: Effective bounds for the zeros of linear recurrences in function fields. J. Theor. Nombres Bordeaux 17(3), 749–766 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fuchs, C., Pethő, A., Tichy, R.F.: On the Diophantine equation \(G_n (x)=G_m (P(x))\). Monatsh. Math. 137, 173–196 (2002)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fuchs, C., Pethő, A., Tichy, R.F.: On the equation \({G_{n}(x)=G_{m}(P(x))}\): Higher order recurrences. Trans. Am. Math. Soc. 355, 4657–4681 (2003)CrossRefGoogle Scholar
  12. 12.
    Fuchs, C., Pethő, A., Tichy, R.F.: On the Diophantine equation \(G_n (x)=G_m (y)\) with \(Q(x,y)=0\). Dev. Math. 16, 199–209. In: Diophantine Approximation Festschrift for Wolfgang Schmidt. (H.P. Schlickewei, K. Schmidt, R.F. Tichy, eds.), Springer-Verlag, Vienna, (2008)Google Scholar
  13. 13.
    Fuchs, C., Zannier, U.: Composite rational functions expressible with few terms. J. Eur. Math. Soc. (JEMS) 14, 175–208 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kirschenhofer, P., Pfeiffer, O.: Diophantine equations between polynomials obeying second order recurrences. Period. Math. Hungar. 47, 119–134 (2003). MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kreso, D.: Diophantine equations in separated variables and lacunary polynomials. Int. J. Number Theory 13, 2055–2074 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kreso, D.: On common values of lacunary polynomials at integer points. N. Y. J. Math. 21, 987–1001 (2015)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Kreso, D., Zieve, M.E.: On factorizations of maps between curves. arXiv:1405.4753
  18. 18.
    Mason, R.C.: Diophantine Equations Over Function Fields. Cambridge University Press, Cambridge (1984)CrossRefGoogle Scholar
  19. 19.
    Müller, P.: Permutation Groups with a cyclic Two-Orbits Subgroup and Monodromy Groups of Siegel Functions, arXiv:math/0110060
  20. 20.
    Müller, P., Zieve, M.E.: On Ritt’s polynomial decomposition theorems. arXiv:0807.3578
  21. 21.
    Ritt, J.F.: Prime and composite polynomials. Trans. Am. Math. Soc. 23, 51–66 (1922)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Schinzel, A.: Polynomials with Special Regard to Reducibility. Cambridge University Press, Cambridge (2000)CrossRefGoogle Scholar
  23. 23.
    Stichtenoth, H.: Function Fields and Codes. Universitext, Springer, Berlin (1993)zbMATHGoogle Scholar
  24. 24.
    Turnwald, G.: On Schur’s conjecture. J. Austral. Math. Soc. Ser. A 58, 312–357 (1995)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Zannier, U.: On the integer solutions of exponential equations in function fields. Ann. Inst. Fourier (Grenoble) 54(4), 849–874 (2004)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Zannier, U.: On the number of terms of a composite polynomial. Acta Arith. 127(2), 157–167 (2007)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Zannier, U.: On composite lacunary polynomials and the proof of a conjecture of Schinzel. Invent. Math. 174, 127–138 (2008)MathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.University of SalzburgSalzburgAustria
  2. 2.Graz University of TechnologyGrazAustria

Personalised recommendations