Comprehensive Systems for Primary Decompositions of Parametric Ideals

We present an effective method for computing parametric primary decomposition via comprehensive Gr\"obner systems. In general, it is very difficult to compute a parametric primary decomposition of a given ideal in the polynomial ring with rational coefficients QA, X where A is the set of parameters and X is the set of ordinary variables. One cause of the difficulty is related to the irreducibility of the specialized polynomial. Thus, we introduce a new notion of ``feasibility'' on the stability of the structure of the ideal in terms of its primary decomposition, and we give a new algorithm for computing a so-called comprehensive system consisting of pairs (C, Q), where for each parameter value in C, the ideal has the stable decomposition Q. We may call this comprehensive system a parametric primary decomposition of the ideal. Also, one can also compute a dense set O such that _ (Q) is a primary decomposition for any C O via irreducible polynomials. In addition, we give several computational examples to examine the effectiveness of our new decomposition.