Monomial Ideals and the Computation of Multiplicities

Abstract

I — (/ii • • • i /m) C R, and let / 6 R. Our main concern is how to carry out the following tests/construction:• Membership Test f E 7 ? • Completeness Test: 1 = 17• Construction Task: I ^ I?• Complexity Cost: cx(J ~> /) ? In the literature one does not find effective methods to generally deal with theseproblems. The difficulty arises, partly, from the specialized nature of the equations the elements need to satisfy. The exception, when we understand the problem fully, is the case of monomial ideals. In this case, / is the monomial ideal defined by the integral convex hull of the exponent vectors of I (see [8, p. 140]). Through the techniques of integer programming, all four problems can, theoretically and often in practice, be solved. For non-monomial ideals, only specialized cases of some of these questions have been dealt with ([7] treats the completeness test for generic complete intersections) .