On the Diophantine equation Gn(x) = Gm(P(x)): Higher-order recurrences

Clemens Fuchs*, Attila Petho, Robert F. Tichy

*Korrespondierende/r Autor/in für diese Arbeit

Publikation: Beitrag in einer FachzeitschriftArtikel

Abstract

Let K be a field of characteristic 0 and let (Gn(x)) n=0 be a linear recurring sequence of degree d in K[x] defined by the initial terms G0, ..., Gd-1 ∈ K[x] and by the difference equation Gn+d(x) = A d-1(x)Gn+d-1(x) + ... + A0(x)Gn(x), for n ≥ 0, with A0, ..., Ad-1 ∈ K [x]. Finally, let P(x) be an element of K[x]. In this paper we are giving fairly general conditions depending only on G0,..., Gd-1 on P, and on A0, ..., Ad-1 under which the Diophantine equation G n(x) = Gm(P(x)) has only finitely many solutions (n, m) ∈ ℤ2, n, m ≥ 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.

Originalspracheenglisch
Seiten (von - bis)4657-4681
Seitenumfang25
FachzeitschriftTransactions of the American Mathematical Society
Jahrgang355
Ausgabenummer11
DOIs
PublikationsstatusVeröffentlicht - 1 Nov 2003

ASJC Scopus subject areas

  • !!Mathematics(all)
  • Angewandte Mathematik

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