  We use the theory of resolutions for a given Hilbert function to investigate
the multiplicity conjectures of Huneke and Srinivasan and Herzog and
Srinivasan. To prove the conjectures for all modules with a particular Hilbert
function, we show that it is enough to prove the statements only for elements
at the bottom of the partially ordered set of resolutions with that Hilbert
function. This enables us to test the conjectured upper bound for the
multiplicity efficiently with the computer algebra system Macaulay 2, and we
verify the upper bound for many Artinian modules in three variables with small
socle degree. Moreover, with this approach, we show that though numerical
techniques have been sufficient in several of the known special cases, they are
insufficient to prove the conjectures in general. Finally, we apply a result of
Herzog and Srinivasan on ideals with a quasipure resolution to prove the upper
bound for Cohen-Macaulay quotients by ideals with generators in high degrees
relative to the regularity.
