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On the representation of integers as linear combinations of consecutive values of a polynomial
Authors:Jacques Boulanger   Jean-Luc Chabert
Affiliation:Department of Mathematics, Université de Picardie, 80039 Amiens, France, LAMFA CNRS-UMR 6140, France ; Department of Mathematics, Université de Picardie, 80039 Amiens, France, LAMFA CNRS-UMR 6140, France
Abstract:Let $K$ be a cyclotomic field with ring of integers $mathcal{O}_{K}$ and let $f$ be a polynomial whose values on $mathbb{Z} $ belong to $mathcal{O}_{K}$. If the ideal of $mathcal{O}_{K}$ generated by the values of $f$ on $mathbb{Z} $ is $mathcal{O}_{K}$ itself, then every algebraic integer $N$ of $K$ may be written in the following form:

begin{displaymath}N=sum_{k=1}^l;varepsilon_{k}f(k)end{displaymath}

for some integer $l$, where the $varepsilon_{k}$'s are roots of unity of $K$. Moreover, there are two effective constants $A$ and $B$ such that the least integer $l$ (for a fixed $N$) is less than $A,widetilde{N}+B$, where

begin{displaymath}widetilde{N}= max_{sigmain Gal(K/mathbb{Q} )} ; vert sigma (N) vert.end{displaymath}

Keywords:
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