ASCM 2007: Computer Mathematics pp 246-262

# A Modified Van der Waerden Algorithm to Decompose Algebraic Varieties and Zero-Dimensional Radical Ideals

• Jia Li
• Xiao-Shan Gao
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5081)

## Abstract

In this paper, we introduce a modified Van der Waerden algorithm to decompose a variety into the union of irreducible varieties. We give an effective representation for irreducible varieties obtained by the algorithm, which allows us to obtain an irredundant decomposition easily. We show that in the zero dimensional case, the polynomial systems for the irreducible varieties obtained in the Van der Waerden algorithm generate prime ideals. As a consequence, we have an algorithm to decompose the radical ideal generated by a finite set of polynomials as the intersection of prime ideals and the degree of the polynomials in the computation is bounded by O(d n ) where d is the degree of the input polynomials and n is the number of variables.

## Keywords

Algebraic variety zero dimensional variety resultant irredundant decomposition

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