Decomposable polynomials in second order linear recurrence sequences

Abstract

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.

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Acknowledgements

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.

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Correspondence to Clemens Fuchs or Dijana Kreso.

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Fuchs, C., Karolus, C. & Kreso, D. Decomposable polynomials in second order linear recurrence sequences. manuscripta math. 159, 321–346 (2019). https://doi.org/10.1007/s00229-018-1070-8

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Mathematics Subject Classification

  • 11B37
  • 11R09
  • 12E99
  • 39B12