Diophantine inequalities with power sums

Scremin, Amedeo

Journal de théorie des nombres de Bordeaux, Tome 19 (2007) no. 2, pp. 547-560.

On appelle somme de puissances toute suite $\alpha :\mathbb{N}\to ℂ$ de nombres complexes de la forme

$$\alpha \left(n\right)=b_1{c}_1^n+b_2{c}_2^n+...+b_h{c}_h^n,$$

où les $b_i\in \overline{\mathbb{Q}}$ et les $c_i\in \mathbb{Z}$ sont fixés. Soit $F(x,y)\in \overline{\mathbb{Q}}[x,y]$ un polynôme unitaire, absolument irréductible, de degré au moins $2$ en $y$. On démontre que les solutions $(n,y)\in \mathbb{N}\times \mathbb{Z}$ de l’inégalité

$$\left|F(\alpha \left(n\right),y)\right|<|\frac{\partial F}{\partial y}(\alpha \left(n\right),y)|·{\left|\alpha \left(n\right)\right|}^{-\epsilon }$$

sont paramétrées par un nombre fini de sommes de puissances. Par conséquent, on déduit la finitude des solutions de l’équation diophantienne

$$F\left(\alpha \right(n),y)=f\left(n\right),$$

où $f\in \mathbb{Z}\left[x\right]$ est un polynôme non constant et $\alpha$ est une somme de puissances non constante.

The ring of power sums is formed by complex functions on $\mathbb{N}$ of the form

$$\alpha \left(n\right)=b_1{c}_1^n+b_2{c}_2^n+...+b_h{c}_h^n,$$

for some $b_i\in \overline{\mathbb{Q}}$ and $c_i\in \mathbb{Z}$. Let $F(x,y)\in \overline{\mathbb{Q}}[x,y]$ be absolutely irreducible, monic and of degree at least $2$ in $y$. We consider Diophantine inequalities of the form

$$\left|F(\alpha \left(n\right),y)\right|<|\frac{\partial F}{\partial y}(\alpha \left(n\right),y)|·{\left|\alpha \left(n\right)\right|}^{-\epsilon }$$

and show that all the solutions $(n,y)\in \mathbb{N}\times \mathbb{Z}$ have $y$ parametrized by some power sums in a finite set. As a consequence, we prove that the equation

$$F\left(\alpha \right(n),y)=f\left(n\right),$$

with $f\in \mathbb{Z}\left[x\right]$ not constant, $F$ monic in $y$ and $\alpha$ not constant, has only finitely many solutions.

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