Determinanti polinomiali-esponenziali

Marcovecchio, Raffaele

Bollettino della Unione matematica italiana, Série 8, 7B (2004) no. 3, pp. 713-730

Résumé

Given $m=2$ or $m=3$ non-zero algebraic numbers $\alpha=(\alpha_{1}, \ldots, \alpha_{m})$ such that $\alpha_{j}/\alpha_{l}$ is not a root of unity for any $j\neq l$ , we consider a class of generalized Vandermonde determinants of order four $G(a; x)$, as $x$ runs over $\mathbb{Z}^{4}$ , involved in some Diophantine problems. We derive an explicit upper bound $N(d)$ for the number of solutions $y\in \mathbb{Z}^{3}$ in general position of the inhomogeneous polynomial-exponential equation $G(a; 0, y)=0$, where the constant $N(d)$ depends only on $d=[\mathbb{Q}(\alpha_{1} , \ldots , \alpha_{m}) : \mathbb{Q}]$.