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

Abstract

Let K K be a cyclotomic field with ring of integers O K \mathcal {O}_{K} and let f f be a polynomial whose values on Z \mathbb {Z} belong to O K \mathcal {O}_{K} . If the ideal of O K \mathcal {O}_{K} generated by the values of f f on Z \mathbb {Z} is O K \mathcal {O}_{K} itself, then every algebraic integer N N of K K may be written in the following form: \[ N = ∑ k = 1 l ε k f ( k ) N=\sum _{k=1}^l\;\varepsilon _{k}f(k) \] for some integer l l , where the ε k \varepsilon _{k} ’s are roots of unity of K K . Moreover, there are two effective constants A A and B B such that the least integer l l (for a fixed N N ) is less than A N ~ + B A\,\widetilde {N}+B , where \[ N ~ = max σ ∈ G a l ( K / Q ) | σ ( N ) | . \widetilde {N}= \max _{\sigma \in Gal(K/\mathbb {Q})} \; \vert \sigma (N) \vert . \]