Computing GCDs of Polynomials over Algebraic Number Fields

Abstract

Modular methods for computing the gcd of two univariate polynomials over an algebraic number field require a priori knowledge about the denominators of the rational numbers in the representation of the gcd. A multiplicative bound for these denominators is derived without assuming that the number generating the field is an algebraic integer. Consequently, the gcd algorithm of Langemyr and McCallum [ J. Symbolic Computation 8, 429 - 448, 1989] can now be applied directly to polynomials that are not necessarily represented in terms of an algebraic integer. Worst-case analyses and experiments with an implementation show that by avoiding a conversion of representation the reduction in computing time can be significant. A further improvement is achieved by using an algorithm for reconstructing a rational number from its modular residue so that the denominator bound need not be explicitly computed. Experiments and analyses suggest that this is a good practical alternative.