Diophantine inequalities with power sums (Q933209)

Let \(\alpha(n)=b_1c_1^n+\cdots+b_hc_h^n\) be a power sum, with \(b_i\) algebraic numbers and \(c_i\) integers and let \(F\) be a monic homogeneous polynomial of degree at least \(2\) with algebraic coefficients. Let \((n,y)\in \mathbb N \times \mathbb Z\) be a solution to the inequality \[ |F(\alpha(n),y)|<|\frac {\partial F}{\partial y}(\alpha (n),y)|\cdot |\alpha(n)|^{-\varepsilon} \] with \(\varepsilon>0\). Then \(y=\beta(n)\), where \(\beta(n)\) is a power sum coming from a finite set of power sums. The author reduces the considered inequality to finitely many inequalities of the type \[ |\tau(n)-y|<e^{-n\varepsilon}, \] with \(\tau\) a power sum. Note that this inequality has been studied by \textit{P. Corvaja} and \textit{U. Zannier} [Indag. Math., New Ser. 9, No. 3, 317--332 (1998; Zbl 0923.11103)]. As a corollary the author can also prove the finiteness of solutions of the following Diophantine equation: \[ F(\alpha(n),y)=f(n), \] where \(f(n)\) is a polynomial.