Regularity of Powers and Products of Ideals

The aim of this thesis is to study the Castelnuovo-Mumford regularity of powers and products of ideals of polynomial rings. In general, it is known that the regularity of large enough powers of ideals are given by a linear function. Moreover, similar result holds for products of large enough powers of ideals which are generated in the same degrees. These are fascinating results, however, there are still missing mysteries on specics of the asymptotic linear functions. For example, the stabilizing index (the smallest power for which the regularity is a linear function) and the so called constant (the constant term in the linear function) is unknown in general. We compute these two missing parts for two deferent classes of ideals using two deferent methods.