#### Glasnik Matematicki, Vol. 38, No.2 (2003), 233-252.

### POLYNOMIAL-EXPONENTIAL EQUATIONS AND LINEAR
RECURRENCES

### Clemens Fuchs

Institut fur Mathematik, Technische Universitat Graz,
Steyrergasse 30, 8010 Graz, Austria

*e-mail:* `clemens.fuchs@tugraz.at`

**Abstract.** Let *K* be an algebraic number field and
let (*G*_{n}) be a linear
recurring sequence defined by

*G*_{n} =
λ_{1} α_{1}^{n} + *P*_{2}(*n*)
α_{2}^{n} + ... +
*P*_{t}(*n*)
α_{t}^{n},

where λ_{1}, α_{1}, ... ,
α_{t}
are non-zero elements of *K* and where
*P*_{i}(*x*)
∈
*K*[*x*] for *i* = 2, ... , *t*.
Furthermore let *f*(*z,x*)
∈
*K*[*z,x*] monic in *x*.
In this paper we want to study the polynomial-exponential Diophantine
equation *f*(*G*_{n}, *x*) = 0.
We want to use a quantitative version of
W. M. Schmidt's Subspace Theorem (due to J.-H. Evertse)
to calculate an upper bound for the number of solutions
(*n,x*) under some additional assumptions.
**2000 Mathematics Subject Classification.**
11D45, 11D61.

**Key words and phrases.** Polynomial-exponential equations,
linear recurrences, Subspace-Theorem.

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