The Hilbert function for any graded module over a field k is defined by the dimension of all of the summands M_b, where 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.
Library of Congress Subject Headings
Characteristic functions; Combinatorial analysis; Algebra, Homological
Department, Program, or Center
School of Mathematical Sciences (COS)
Barouti, Maria, "Computing Hilbert Functions using the Syzygy and LCM-lattice methods" (2011). Thesis. Rochester Institute of Technology. Accessed from
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