Result: Norms of factors of polynomials

Let \(A(x)\in\mathbb{Z}[x]\) be a nonzero polynomial of degree \(d\). Denote the Euclidean (or \(\ell^2\)) norm of \(A\) by \(\| A\|\). The main result of the paper is the following: Theorem 1. Let \(A(x)\in\mathbb{Z}[x]\) be a polynomial having no cyclotomic factors. Let \(N \geq 1\). If \(Q(x) \in\mathbb{Z}[x]\) and \(\| AQ\|\leq N\), then \(\| Q\|\) is bounded by a function depending only on \(A(x)\) and \(N\). Note that this bound on \(\| Q\| \) does not depend on the degree of \(Q\). The restriction that \(A\) have no cyclotomic factors is necessary as the authors show by simple examples. Also, the bound must depend on \(A(x)\) and not simply on \(N\) as shown by further examples. The proof of Theorem 1 is quite intricate and the bound on \(\| Q\|\), although explicit, can be very large. A simpler proof and smaller bound is possible if one assumes that \(A(x)\) has no zeros on the unit circle, as shown in Theorem 3. A second related question considered in the paper is to determine a non-zero multiple \(P(x)=A(x)Q(x)\) of a given \(A(x)\) that has the smallest Euclidean norm. Theorem 2 shows that if \(A(x)\) has no cyclotomic factors, then the set of such \(P\) is finite and can be effectively determined. This generalizes an earlier result of \textit{M. Filaseta, M. Robinson} and \textit{F. Wheeler} [J. Algorithms 16, 309-333 (1994; Zbl 0795.11064)] which made the additional assumption that \(A(x)\) be irreducible.