The Minimal Euclidean Norm of an Algebraic Number Is Effectively Computable

Abstract

For P ∈ Z [ x], let || P|| denote the Euclidean norm of the coefficient vector of P. For an algebraic number α, with minimal polynomial A over Z , define the Euclidean norm of α by ||α|| = || A||. Define the minimal Euclidean norm of α by ||α|| min = min{|| P|| : P ∈ Z [ x], P(α) = 0, P ≢ 0}. Given an algebraic number α, we show there exists a P ∈ 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 Z is a root of unity if and only if the set { QA : Q ∈ 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.