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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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