A Survey of Primary Decomposition using GrSbner Bases

A Survey of Primary Decomposition using GrSbner Bases MICHELLE WILSON Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Master of Science We present a survey of primary decomposition of ideals in a noetherian commutative polynomial ring R[x] = R[xi,..., x,]. With the use of ideal operations introduced and the lexicographical Gr6bner bases of ideals in R[x], we show that it is possible to compute a primary decomposition of these ideals. Our method involves the reduction of the general primary decomposition problem to the case of zero-dimensional ideals. Furthermore, we solve the general primary decomposition problem when the coefficient ring is a principal ideal domain. For the zero-dimensional ideals in R[x], we compute inductively their irredundant primary decomposition. In addition, we show that we can compute primary decomposition of zero-dimensional ideals over a field of characteristic zero. We do this by considering ideals in "general position". Finally, we present algorithms to perform the computation of primary decomposition in the cases discussed. Thesis Advisor: Dr. Steven Kleiman Title: Professor of Pure Mathematics