Using Interval Computation with the Mahler Measure for Zero Determination of Algebraic Numbers
Hiroshi Sekigawa · Institutional Repositories DataBase (IRDB) · 1998
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.