Improvements of the Power-series Coefficient Polynomial Remainder Sequence GCD algorithm

Abstract

The PC-PRS (Power-series Coefficient Polynomial Remainder Sequence) GCD is a multivariate polynomial GCD algorithm constructed recently, and is quite practical. The algorithm calculates a PRS by treating coefficients w.r.t. the main variable as a truncated power-series. Efficiency of the PC-PRS GCD algorithm depends greatly on how the power-series is truncated, and the algorithm employs a degree bound of GCD, determined theoretically. This theoretical bound is often larger than the true bound of GCD. In the first part of this paper, a practical bound is introduced so as to improve the computing time of the original algorithm. The practical bound is, however, not always large enough to calculate the true GCD. When the practical bound is small, the construction methods are applied to lift the truncated polynomials to the true GCD. In the remaining parts, the improved algorithm is combined with three construction methods; one is the Hensel construction, and the other two use the extended Euclidean relation and the division relation. Empirical tests show that our algorithms are efficient for polynomials of low to middle degree and many variables.