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On polynomial-exponential equations. (English) Zbl 0805.11029
Let \(K\) be an algebraic number field and \(S\) a finite set of places on \(K\) containing the archimedean places. For \(l=1, \dots,k\), let \(P_ l \in K[X_ 1, \dots, X_ n]\) be a polynomial and \(\underline \alpha_ l = (\alpha_{l1}, \dots, \alpha_{ln})\) a vector of \(S\)-units. The authors consider the polynomial-exponential equation \[ \sum^ k_{l=1} P_ l({\mathbf x}) \underline \alpha_ l^{{\mathbf x}} = 0, \] where \(\underline \alpha^{\mathbf x}_ l = \alpha_{l1}^{x_ 1} \dots \alpha_{ln}^{x_ n}\). For a partition \({\mathcal P} = \{P_ 1, \dots, P_ t\}\) of \(\{1,\dots,k\}\), denote by \({\mathfrak S} ({\mathcal P})\) the set of solutions of \((*)\) with \(\sum_{l \in P_ j} P_ l ({\mathbf x}) \underline \alpha_ l^{{\mathbf x}} = 0\) for \(j=1, \dots,t\) and \(\sum_{l \in P} P_ l({\mathbf x}) \underline \alpha_ l^{{\mathbf x}} \neq 0\) for each non-empty set \(P\) properly contained in one of the sets of \({\mathcal P}\). Further, let \(G({\mathcal P})\) be the abelian group of \({\mathbf x} \in \mathbb{Z}^ n\) such that \(\underline \alpha_ l^{\mathbf x} = \underline \alpha^{\mathbf x}_ m\) for each pair \(l,m\) belonging to the same set of \({\mathcal P}\). A special case of a theorem of M. Laurent [J. Number Theory 31, 24-53 (1989; Zbl 0661.10027)] states that if \(G({\mathcal P}) = \{\text{\textbf{0}}\}\), then \({\mathfrak S} ({\mathcal P})\) is finite. The authors show that in this case, \({\mathfrak S} ({\mathcal P})\) has cardinality at most \(2^ c\), where \(c=20n^ 4 + ns^ 7 \cdot 2^{43d!(Dk)!}\), with \(s\) denoting the cardinality of \(S\), \(d\) the degree of \(K\) and \(D={n + \delta \choose \delta}\), \(\delta\) being an upper bound for the total degrees of the polynomials \(P_ l\).
In the proof, the authors use Schlickewei’s \(p\)-adic generalisation of Schmidt’s quantitative Subspace Theorem. The authors use an ingenious determinant argument to get rid of the dependence on the coefficients of the polynomial \(P_ l\).

MSC:
11D61 Exponential Diophantine equations
11D72 Diophantine equations in many variables
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References:
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