The Minimal Euclidean Norm of an Algebraic Number Is Effectively Computable

Abstract

For P ∈ $\text{Z}$[x], let ||P|| denote the Euclidean norm of the coefficient vector of P. For an algebraic number α, with minimal polynomial A over $\text{Z}$, define the Euclidean norm of α by ||α|| = ||A||. Define the minimal Euclidean norm of α by ||α||_min = min{||P|| : P ∈ $\text{Z}$[x], P(α) = 0, P ≢ 0}. Given an algebraic number α, we show there exists a P ∈ $\text{Z}$[x] with P(α) = 0 and ||P|| = ||α||_min such that the degree of P is bounded above by an explicit function of deg α, ||α||, and ||α||_min. As a result, we are able to prove that both P and ||α||_min can be effectively computed using a suitable search procedure. As an indication of the difficulties involved, we show that the determination of P is equivalent to finding a shortest nonzero vector in an infinite union of certain lattices. After introducing several techniques for reducing the search space, a practical algorithm is presented that has been successful in computing ||α||_min, provided the degree and Euclidean norm of α are both sufficiently small. We also obtain the following unusual characterization of the roots of unity: An algebraic number a with minimal polynomial A over $\text{Z}$ is a root of unity if and only if the set {QA : Q ∈ $\text{Z}$[x], Q(0) ≠ 0, ||QA|| = ||α||_min} contains infinitely many polynomials. We show how to extend the above results to other l_p norms. Some related open problems are also discussed.