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

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- by Jacques Boulanger and Jean-Luc Chabert PDF
- Trans. Amer. Math. Soc.
**356**(2004), 5071-5088 Request permission

## 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: \[ N=\sum _{k=1}^l\;\varepsilon _{k}f(k)\] 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 \[ \widetilde {N}= \max _{\sigma \in Gal(K/\mathbb {Q})} \; \vert \sigma (N) \vert .\]## References

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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
- Received by editor(s): April 20, 2003
- Received by editor(s) in revised form: September 24, 2003
- Published electronically: June 29, 2004
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**356**(2004), 5071-5088 - MSC (2000): Primary 11A67; Secondary 11P05, 11R18, 13F20
- DOI: https://doi.org/10.1090/S0002-9947-04-03569-X
- MathSciNet review: 2084411