Multiplicity-preserving triangular set decomposition of two polynomials

Abstract

In this paper, a multiplicity-preserving triangular set decomposition algorithm is proposed for a system of two polynomials, which involves only computing the primitive polynomial remainder sequence of two polynomials once and certain GCD computations. The algorithm decomposes the unmixed variety defined by two polynomials into square free and disjoint (for non-vertical components, see Definition 4) algebraic cycles represented by triangular sets, which may have negative multiplicities. Thus, the authors can count the multiplicities of the non-vertical components. In the bivariate case, the authors give a complete algorithm to decompose the system into zeros represented by triangular sets with multiplicities. The authors also analyze the complexity of the algorithm in the bivariate case. The authors implement the algorithm and show the effectiveness of the method with extensive