Student Commutative Algebra Seminar: The Generic Initial Ideal

Given an ideal J in a polynomial ring R = K[x_1, ..., x_n], the initial ideal of J -- the ideal generated by the leading monomial terms in a Gröbner basis for J -- is a useful combinatorial invariant. If you perform a generic change of coordinates on R before computing the initial ideal of J, the resulting initial ideal turns out to be especially well-behaved. We call this ideal the generic initial ideal of J. The generic change of variables eliminates the dependence on the original coordinates and yields an initial ideal of minimal complexity.

We'll start by reviewing the basics of monomial orders and Gröbner bases. Next, we'll develop the theory of the generic initial ideal, proving its existence and summarizing basic properties. Lastly, we'll showcase applications of the generic initial ideal in commutative algebra and algebraic geometry.