Sum-and-quotient characteristic decomposition of polynomial ideals

This paper studies the algorithmic decomposition of polynomial ideals using the sum-and-quotient operation, a key technique implicitly involved in the process of computing Hilbert polynomials. We extend the framework of characteristic decomposition algorithms that maintain Hilbert polynomial relationships to the positive-dimensional case. In particular, we propose algorithms that successively apply the sum-and-quotient lemma to decompose any given polynomial ideal into finitely many ideals generated by regular sets or regular sequences, such that certain relations among the zero sets and the Hilbert polynomials are simultaneously preserved. This approach offers a new framework for representing the zero sets of polynomial ideals with multiplicities and reveals inherent connections among key concepts in the algorithmic theories of triangular sets, Gröbner bases, and Hilbert polynomials. We provide examples to illustrate computational properties and contrasts between our method and pseudo-division-based triangular decomposition algorithms. Experimental results demonstrate that the performance of our method is comparable with the existing triangular decomposition algorithms based solely on Gröbner bases computation.