TY - JOUR
T1 - On the Diophantine equation Gn(x) = Gm(P(x))
T2 - Higher-order recurrences
AU - Fuchs, Clemens
AU - Petho, Attila
AU - Tichy, Robert F.
PY - 2003/11/1
Y1 - 2003/11/1
N2 - 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.
AB - 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.
KW - Diophantine equations
KW - Linear recurring sequences
KW - S-unit equations
UR - http://www.scopus.com/inward/record.url?scp=0242350486&partnerID=8YFLogxK
U2 - 10.1090/S0002-9947-03-03325-7
DO - 10.1090/S0002-9947-03-03325-7
M3 - Article
AN - SCOPUS:0242350486
SN - 0002-9947
VL - 355
SP - 4657
EP - 4681
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
IS - 11
ER -