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Computing strong regular characteristic pairs with Groebner bases

The W-characteristic set of a polynomial ideal is the minimal triangular set contained in the reduced lexicographical Groebner basis of the ideal. A pair (G,C) of polynomial sets is a strong regular characteristic pair if G is a reduced lexicographical Groebner basis, C is the W-characteristic set of the ideal <G>, the saturated ideal sat(C) of C is equal to <G>, and C is regular. In this paper, we show that for any polynomial ideal I with given generators one can either detect that I is unit, or construct a strong regular characteristic pair (G,C) by computing Groebner bases such that I$\subseteq$sat(C)=<G> and sat(C) divides I, so the ideal I can be split into the saturated ideal sat(C) and the quotient ideal I:sat(C). Based on this strategy of splitting by means of quotient and with Groebner basis and ideal computations, we devise a simple algorithm to decompose an arbitrary polynomial set F into finitely many strong regular characteristic pairs, from which two representations for the zeros of F are obtained: one in terms of strong regular Groebner bases and the other in terms of regular triangular sets. We present some properties about strong regular characteristic pairs and characteristic decomposition and illustrate the proposed algorithm and its performance by examples and experimental results.