The Intersect command computes the intersection of two or more polynomial ideals.  The set of variables is extended to include the variables of each ideal.  If the ideals cannot be put into a common polynomial ring, then an error is produced.

The Intersect command makes extensive use of generators that are common to one or more of the ideals.  Thus for certain applications, such as intersecting prime components, avoid using the Simplify command prior to using Intersect.

Cox, D.; Little, J.; and O'Shea, D. Ideals, Varieties, and Algorithms. 2nd ed. New York: Springer-Verlag, 1997.