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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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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 \[ \widetilde {N}= \max _{\sigma \in Gal(K/\mathbb {Q})} \; \vert \sigma (N) \vert .\]
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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