In the study of ideals in polynomial rings over a field, one can glean a great deal of the properties and the invariants of a homogeneous ideal from its minimal graded free resolution. By a classical result of David Hilbert, which dates back to the nineteenth century, these resolutions are always finite and their length is bounded above by the dimension of the ring. In this thesis we consider the question whether the projective dimension (or equivalently, the Castelnuovo-Mumford regularity) of an ideal can be bounded solely in terms of its number of generators and the degrees of those generators, but independently of (the dimension of) the ring. We give an affirmative answer in the case of ideals generated by three cubic forms by showing that the projective dimension of three cubits (in any polynomial ring over a field) is at most 36. We also settle the open question of whether three cubic forms can have projective dimension greater than 4 by constructing an example with projective dimension equal to 5.
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