Diophantine inequalities with power sums

Scremin, Amedeo. "Diophantine inequalities with power sums." Journal de Théorie des Nombres de Bordeaux 19.2 (2007): 547-560. <http://eudml.org/doc/249949>.

@article{Scremin2007,
abstract = {The ring of power sums is formed by complex functions on $\mathbb\{N\}$ of the form\[\alpha (n) = b\_1 c\_1^n + b\_2 c\_2^n + \ldots + b\_h c\_h^n,\]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\[|F(\alpha (n),y)| &lt; \Big | \frac\{\partial F\}\{\partial y\} (\alpha (n),y)\Big | \cdot |\alpha (n)|^\{-\varepsilon \}\]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\[F(\alpha (n),y)=f(n),\]with $f\in \mathbb\{Z\}[x]$ not constant, $F$ monic in $y$ and $\alpha $ not constant, has only finitely many solutions.},
affiliation = {Institut für Mathematik A Technische Universität Graz Steyrergasse 30 A-8010 Graz, Austria},
author = {Scremin, Amedeo},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Diophantine inequalities; power sums},
language = {eng},
number = {2},
pages = {547-560},
publisher = {Université Bordeaux 1},
title = {Diophantine inequalities with power sums},
url = {http://eudml.org/doc/249949},
volume = {19},
year = {2007},
}

TY - JOUR
AU - Scremin, Amedeo
TI - Diophantine inequalities with power sums
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2007
PB - Université Bordeaux 1
VL - 19
IS - 2
SP - 547
EP - 560
AB - The ring of power sums is formed by complex functions on $\mathbb{N}$ of the form\[\alpha (n) = b_1 c_1^n + b_2 c_2^n + \ldots + b_h c_h^n,\]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\[|F(\alpha (n),y)| &lt; \Big | \frac{\partial F}{\partial y} (\alpha (n),y)\Big | \cdot |\alpha (n)|^{-\varepsilon }\]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\[F(\alpha (n),y)=f(n),\]with $f\in \mathbb{Z}[x]$ not constant, $F$ monic in $y$ and $\alpha $ not constant, has only finitely many solutions.
LA - eng
KW - Diophantine inequalities; power sums
UR - http://eudml.org/doc/249949
ER -