Computations in Residue Class Rings of (Parametric) Polynomial Ideals

The paper proposes a new construction for studying various properties of a polynomial with respect to an ideal generated by a finite basis of polynomials. These properties include, among others, (i) checking whether the polynomial is a zero divisor in the residue class ring defined by the associated ideal and (ii) checking whether the polynomial is invertible in the residue class ring defined by the associated ideal. Further it can be decided whether the polynomial is in the radical of the ideal. Some of the byproducts of this construction are that it is possible to be more discriminatory in determining whether the polynomial is a zero divisor (invertible, respectively) in the quotient ring defined by the ideal, or the colon ideal constructed by localization using the polynomial. The method of course also computes the smallest integer which gives the saturation ideal of the ideal with respect to the polynomial. The construction uses a single Gröbner basis computation to achieve all these results. The construction carries over naturally to parametric ideals, where all these properties of a parametric polynomial can be determined under various parameter specialization using an algorithm for computing a (minimal) comprehensive Gröbner system of a parametric ideal. The work has been motivated by the application of geometric theorem discovery and proving. Many geometry problems have been solved by the proposed method.