Direct methods for primary decomposition

Let I be an ideal in a polynomial ring over a perfect field. We give new methods for comput ing the equidimensional parts and radical of I, for localizing I with respect to another ideal, and thus for finding the pr imary decomposit ion of / . Our methods rest on modern ideas from commutat ive 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).