Decomposition of Polynomial Ideals into Triangular Regular Sequences

This paper studies the representation of the set of zeros with multiplicities for an ideal generated by a given set of multivariate polynomials in terms of triangular regular sequences, whose dimensions and degrees can be read out directly. A new algebro-geometric approach is proposed that enables one to decompose any polynomial ideal into finitely many triangular regular sequences of polynomials such that certain implicit relations between the Hilbert polynomials and explicit relations between the sets of zeros of the ideals generated by the regular sequences are preserved. The decomposition algorithms make use of the properties and computations of W-characteristic sets of polynomial ideals and perform simultaneous sum-and-quotient operation, a key technique that is used implicitly in the recursive process of computing Hilbert polynomials. The present work elaborates and reveals inherent connections between some commonly used concepts in the algorithmic theories of triangular sets, Gröbner bases, and Hilbert polynomials. Examples are provided to illustrate the computational aspects and differences of our approach from that of pseudo-division-based triangular decomposition.