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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
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