Simple Characteristic Decomposition of Polynomial Sets

Characteristic pairs consist of lexicographical Gröbner bases and the minimal triangular sets, called W-characteristic sets, contained in them, and they are good representations of multivariate polynomial ideals in terms of Gröbner bases and triangular sets simultaneously. In this paper, it is studied how to decompose a polynomial set of arbitrary dimensions into characteristic pairs with simple W-characteristic sets, and two algorithms are proposed over fields of characteristic zero and over finite fields respectively. Both of the algorithms rely on the concept of strong regular characteristic divisors, and the one for fields of characteristic zero also uses Lazard lemma to test whether an ideal is radical. Experimental results are presented to illustrate the effectiveness of the proposed algorithms.