The Minimal Euclidean Norm of an Algebraic Number Is Effectively Computable

For P 2 Z[x], let kPk denote the Euclidean norm of the coe cient vector of P . For an algebraic number , with minimal polynomial A, de ne the Euclidean norm of by k k = kkAk ; where k is the smallest positive integer for which kA 2 Z[x]. De ne the minimal Euclidean norm of by k k min = min kPk : P 2Z[x]; P ( ) = 0; P 6 0 : Given an algebraic number , we show there exists a P 2 Z[x] with P ( ) = 0 and kPk = k k min such that the degree of P is bounded above by an explicit function of deg , k k, and k k min . As a result, we are able to prove that both P and k k min can be e ectively computed using a suitable search procedure. As an indication of the di culties involved, we show that the determination of P is equivalent to nding a shortest nonzero vector in an in nite union of certain lattices. After introducing several techniques for reducing the search space, a practical algorithm is presented which has been successful in computing k k min provided the degree and Euclidean norm of are both su ciently small. We also obtain the following unusual characterization of the roots of unity: An algebraic number is a root of unity if and only if the set P : P 2Z[x]; P ( ) = 0; P (0) 6= 0; kPk = k k min contains in nitely many polynomials. We show how to extend the above results to other lp norms. Some related open problems are also discussed.