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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the representation of integers as linear combinations of consecutive values of a polynomial


Authors: Jacques Boulanger and Jean-Luc Chabert
Journal: Trans. Amer. Math. Soc. 356 (2004), 5071-5088
MSC (2000): Primary 11A67; Secondary 11P05, 11R18, 13F20
Published electronically: June 29, 2004
MathSciNet review: 2084411
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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_{\sigma\in Gal(K/\mathbb{Q} )} \; \vert \sigma (N) \vert.\end{displaymath}


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Additional Information

Jacques Boulanger
Affiliation: Department of Mathematics, Université de Picardie, 80039 Amiens, France, LAMFA CNRS-UMR 6140, France
Email: jaboulanger@wanadoo.fr

Jean-Luc Chabert
Affiliation: Department of Mathematics, Université de Picardie, 80039 Amiens, France, LAMFA CNRS-UMR 6140, France
Email: jean-luc.chabert@u-picardie.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03569-X
PII: S 0002-9947(04)03569-X
Received by editor(s): April 20, 2003
Received by editor(s) in revised form: September 24, 2003
Published electronically: June 29, 2004
Article copyright: © Copyright 2004 American Mathematical Society