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

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