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

Authors:
Jia Li;Xiao-Shan Gao
Affiliations:
Key Laboratory of Mathematics Mechanization Institute of Systems Science, AMSS, Chinese Academy of Sciences, Beijing, China 100080;Key Laboratory of Mathematics Mechanization Institute of Systems Science, AMSS, Chinese Academy of Sciences, Beijing, China 100080
Venue:
Computer Mathematics
Year:
2008

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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(dn) where dis the degree of the input polynomials and nis the number of variables.