## The set of solutions of a polynomial-exponential equation.(English)Zbl 0930.11018

Set $$\Lambda=1,2,\dots,k$$. When $$P$$ is a partition of $$\Lambda$$, we will write $$\lambda\in P$$ to mean that $$\lambda$$ is one of the subsets of $$\Lambda$$ appearing in $$P$$. In this paper the author considers the system of equations. $\sum_{\ell\in\lambda}P_\ell(x)\alpha^x_l=0\qquad (\lambda\in P),\tag{1}$ in the variable $${\mathbf x}=(x_1,x_2,\dots,x_n)\in\mathbb{Z}^n$$, where $$P_\ell$$ are polynomials with coefficients in a number field $$K$$ and $$\alpha^{\mathbf x}=\alpha^{x_1}_{l1}\cdots\alpha_{ln}^{x_n}$$ with $$\alpha_{ij}\in K^{\mathbf x}$$ $$(1\leq l\leq k,\;1\leq j\leq n)$$. Let $$G(P)$$ be the subgroup of $$\mathbb{Z}^n$$ consisting of $${\mathbf x}$$ such that $$\alpha_l^{\mathbf x}=\alpha_m^{\mathbf x}$$ whenever $$\ell$$ and $$m$$ lie in same set $$\lambda$$ of $$P$$.
The author investigates the number of solutions of (1) when $$G(P)\neq\{0\}$$.
Reviewer: D.Acu (Sibiu)

### MSC:

 11D61 Exponential Diophantine equations

### Keywords:

exponential diophantine equations
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