  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.
