Decomposable polynomials in second order linear recurrence sequences View Full Text


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Article Info

DATE

2018-10-04

AUTHORS

Clemens Fuchs, Christina Karolus, Dijana Kreso

ABSTRACT

We study elements of second order linear recurrence sequences (Gn)n=0∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(G_n)_{n= 0}^{\infty }$$\end{document} of polynomials in C[x]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {C}}}[x]$$\end{document} which are decomposable, i.e. representable as Gn=g∘h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_n=g\circ h$$\end{document} for some g,h∈C[x]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g, h\in {{\mathbb {C}}}[x]$$\end{document} satisfying degg,degh>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\deg g,\deg h>1$$\end{document}. Under certain assumptions, and provided that h is not of particular type, we show that degg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\deg g$$\end{document} may be bounded by a constant independent of n, depending only on the sequence. More... »

PAGES

321-346

References to SciGraph publications

  • 2003-09. Diophantine equations between polynomials obeying second order recurrences in PERIODICA MATHEMATICA HUNGARICA
  • 2009-11. Polynomials with a common composite in ISRAEL JOURNAL OF MATHEMATICS
  • 2008-04-24. On composite lacunary polynomials and the proof of a conjecture of Schinzel in INVENTIONES MATHEMATICAE
  • 2002-11. On the Diophantine Equation Gn(x) = Gm(P(x)) in MONATSHEFTE FÜR MATHEMATIK
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    http://scigraph.springernature.com/pub.10.1007/s00229-018-1070-8

    DOI

    http://dx.doi.org/10.1007/s00229-018-1070-8

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