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Transactions of the American Mathematical Society

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On the Diophantine equation $G_n(x)=G_m(P(x))$: Higher-order recurrences

Authors: Clemens Fuchs, Attila Petho and Robert F. Tichy
Journal: Trans. Amer. Math. Soc. 355 (2003), 4657-4681
MSC (2000): Primary 11D45; Secondary 11D04, 11D61, 11B37
Published electronically: June 10, 2003
MathSciNet review: 1990766
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $\mathbf{K}$ be a field of characteristic $0$ and let $(G_{n}(x))_{n=0}^{\infty}$ be a linear recurring sequence of degree $d$ in $\mathbf{K}[x]$ defined by the initial terms $G_0,\ldots,G_{d-1}\in\mathbf{K}[x]$ and by the difference equation

\begin{displaymath}G_{n+d}(x)=A_{d-1}(x)G_{n+d-1}(x)+\cdots+A_0(x)G_{n}(x), \quad \mbox{for} \,\, n\geq 0,\end{displaymath}

with $A_0,\ldots,A_{d-1}\in\mathbf{K}[x]$. Finally, let $P(x)$ be an element of $\mathbf{K}[x]$. In this paper we are giving fairly general conditions depending only on $G_0,\ldots,G_{d-1},$ on $P$, and on $A_0,\ldots,A_{d-1}$ under which the Diophantine equation


has only finitely many solutions $(n,m)\in \mathbb{Z}^{2},n,m\geq 0$. Moreover, we are giving an upper bound for the number of solutions, which depends only on $d$. This paper is a continuation of the work of the authors on this equation in the case of second-order linear recurring sequences.

References [Enhancements On Off] (What's this?)

  • 1. Alan Baker (ed.), New advances in transcendence theory, Cambridge University Press, Cambridge, 1988. MR 971989
  • 2. W. D. Brownawell and D. W. Masser, Vanishing sums in function fields, Math. Proc. Cambridge Philos. Soc. 100 (1986), no. 3, 427–434. MR 857720, 10.1017/S0305004100066184
  • 3. F. Beukers and H. P. Schlickewei, The equation 𝑥+𝑦=1 in finitely generated groups, Acta Arith. 78 (1996), no. 2, 189–199. MR 1424539
  • 4. Jan-Hendrik Evertse, On sums of 𝑆-units and linear recurrences, Compositio Math. 53 (1984), no. 2, 225–244. MR 766298
  • 5. Jan-Hendrik Evertse, On equations in two 𝑆-units over function fields of characteristic 0, Acta Arith. 47 (1986), no. 3, 233–253. MR 870667
  • 6. J.-H. Evertse and K. Győry, On the numbers of solutions of weighted unit equations, Compositio Math. 66 (1988), no. 3, 329–354. MR 948309
  • 7. J.-H. Evertse, K. Győry, C. L. Stewart, and R. Tijdeman, 𝑆-unit equations and their applications, New advances in transcendence theory (Durham, 1986) Cambridge Univ. Press, Cambridge, 1988, pp. 110–174. MR 971998
  • 8. Jan-Hendrik Evertse and Hans Peter Schlickewei, The absolute subspace theorem and linear equations with unknowns from a multiplicative group, Number theory in progress, Vol. 1 (Zakopane-Kościelisko, 1997) de Gruyter, Berlin, 1999, pp. 121–142. MR 1689503
  • 9. J.-H. EVERTSE, H. P. SCHLICKEWEI AND W. M. SCHMIDT, Linear equations in variables which lie in a multiplicative group, Ann. Math. 155 (2002), 807-836.
  • 10. C. FUCHS, Quantitative finiteness results for Diophantine equations, Ph.D. thesis, TU Graz (2002).
  • 11. C. FUCHS, On the equation $G_n(x)=G_m(P(x))$ for third order linear recurring sequences, Portugal. Math., to appear.
  • 12. C. FUCHS, A. PETHSO AND R. F. TICHY, On the Diophantine equation $G_{n}(x)=G_{m}(P(x))$, Monatsh. Math. 137 (2002), 173-196.
  • 13. Christer Lech, A note on recurring series, Ark. Mat. 2 (1953), 417–421. MR 0056634
  • 14. A. J. VAN DER POORTEN AND H. P. SCHLICKEWEI, The growth conditions for recurrence sequences, Macquarie Univ. Math. Rep. 82-0041, North Ryde, Australia, 1982.
  • 15. A. Schinzel, Reducibility of polynomials in several variables, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 11 (1963), 633–638. MR 0159816
  • 16. A. Schinzel, Polynomials with special regard to reducibility, Encyclopedia of Mathematics and its Applications, vol. 77, Cambridge University Press, Cambridge, 2000. With an appendix by Umberto Zannier. MR 1770638
  • 17. Hans Peter Schlickewei, The multiplicity of binary recurrences, Invent. Math. 129 (1997), no. 1, 11–36. MR 1464864, 10.1007/s002220050156
  • 18. Wolfgang M. Schmidt, The zero multiplicity of linear recurrence sequences, Acta Math. 182 (1999), no. 2, 243–282. MR 1710183, 10.1007/BF02392575
  • 19. Henning Stichtenoth, Algebraic function fields and codes, Universitext, Springer-Verlag, Berlin, 1993. MR 1251961

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Additional Information

Clemens Fuchs
Affiliation: Institut für Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria

Attila Petho
Affiliation: Institute for Mathematics and Informatics, University of Debrecen, H-4010 Debrecen, PO Box 12, Hungary

Robert F. Tichy
Affiliation: Institut für Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria

Keywords: Diophantine equations, linear recurring sequences, $S$-unit equations
Received by editor(s): October 18, 2002
Received by editor(s) in revised form: February 7, 2003
Published electronically: June 10, 2003
Additional Notes: This work was supported by the Austrian Science Foundation FWF, grant S8307-MAT
The second author was supported by the Hungarian National Foundation for Scientific Research, Grant Nos. 29330 and 38225
Dedicated: Dedicated to Wolfgang M. Schmidt on the occasion of his 70th birthday.
Article copyright: © Copyright 2003 American Mathematical Society