Direct methods for primary decomposition

TL;DR: In this article, the authors have given new methods for computing the equidimensional parts and radical of an ideal in a polynomial ring over a perfect field, for localizing the ideal with respect to another ideal, and for finding the primary decomposition of the ideal.

Abstract: LetI be an ideal in a polynomial ring over a perfect field. We given new methods for computing the equidimensional parts and radical ofI, for localizingI with respect to another ideal, and thus for finding the primary decomposition ofI. Our methods rest on modern ideas from commutative algebra, and are direct in the sense that they avoid the generic projections used by Hermann (1926) and all others until now. Some of our methods are practical for certain classes of interesting problems, and have been implemented in the computer algebra system Macaulay of Bayer and Stillman (1982–1992).