Computing Hilbert Functions using the Syzygy and LCM-lattice methods.

摘要

The Hilbert function for any graded module M = ⨁i∈N Mi over a field k is defined by HFM,b =dimk Mb, where integer b indicates the graded component being considered.;One standard approach to computing the Hilbert function is to come up with a free-resolution for the graded module M and another is via a Hilbert power series which serves as a generating function. Using combinatorics and homological algebra we develop three alternative ways to generate the values of a Hilbert function when the graded module is a quotient ring over a field. Two of these approaches (which we've called the lcm-Lattice method and the Syzygy method) are conceptually combinatorial and work for any polynomial quotient ring over a field. The third approach, which we call the Hilbert function table method, also uses syzygies but the approach is better described in terms of homological algebra.