Chapter 13 - Algebraic factorization and GCD computation

Abstract

This chapter describes several algorithms for factorization and greatest common divisor (GCD) computation of polynomials over algebraic extension fields. These algorithms are common in using the method of characteristic sets, and have found encouraging applications to geometric theorem proving, irreducible decomposition of algebraic varieties, and other problems. Some information about implementations and performance of the presented algorithms is provided. A method using transformation and triangularization is also discussed. The basic idea underlying the method is the reduction of polynomial factorization over algebraic extension fields to that over the rational number field via linear transformation, and the computation of characteristic sets with respect to a proper variable ordering. The factors over the algebraic extension fields are finally determined through algebraic GCD computation. It is found that an immediate variation of the algorithm is to compute not only the characteristic set but also the characteristic series. The irreducible factors are determined from those ascending sets in the series whose irreducibility can easily be verified. The hybrid method with modular techniques is also elaborated.