# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## On the representation of integers as linear combinations of consecutive values of a polynomialHTML articles powered by AMS MathViewer

by Jacques Boulanger and Jean-Luc Chabert
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 $\widetilde {N}= \max _{\sigma \in Gal(K/\mathbb {Q})} \; \vert \sigma (N) \vert .$
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• Jacques Boulanger
• Affiliation: Department of Mathematics, Université de Picardie, 80039 Amiens, France, LAMFA CNRS-UMR 6140, France