Using Interval Computation with the Mahler Measure for Zero Determination of Algebraic Numbers

We propose a new zero determination principle for an algebraic number α obtained after performing ring operations among algebraic numbers α1，...，αn. We assume that each αi is represented by its minimal polynomial over Q and its approximate value as an interval that contains only αi among its conjugates. The principle of zero determination is as follows： by the estimate of the Mahler measure for α and by the interval for α，we can correctly determine whether α is zero or not with a finite precision value of approximation.We propose two practical usages of the principle. One method computes both intervals and the Mahler measures simultaneously. The other method utilizes a history of computation to compute the Mahler measures only when they are required. Furthermore，we sharpen inequalities on the Mahler measure after ring operations among algebraic numbers.Proceedings of 2nd Risa Consortium at Ehime University 6-9 March, 1997/ edited by Matu-Tarow Noda Kiyoko Nishizawa Tomokatsu Sait