Integral Closure Presentations and Membership

Algebra Seminar
Tuesday, February 7, 2012 - 14:00
1 hour (actually 50 minutes)
Skiles 006
Auburn University
Let I be an ideal in a polynomial ring R := F[x_n,...,x_1] Let A := R/I be the corresponding quotient ring, and let Q(A) be its eld of fractions. The integral closure C(A, Q(A)) of A in Q(A) is a subring of the latter. But it is often given as a separate quotient ring, a presentation. Surprisingly, different computer algebra systems (Magma, Macaulay2, and Singular) choose to produce very different presentations. Some of these opt for presentations that have seductive forms, but miss the most important, namely a form that allows for determining when elements ofQ(A) are in C(A,Q(A)). This is called membership and is directly related to determining isomorphism.