  There is a longstanding conjecture by Fr\"oberg about the Hilbert series of
the ring $R/I$, where $R$ is a polynomial ring, and $I$ an ideal generated by
generic forms. We prove this conjecture true in the case when $I$ is generated
by a large number of forms, all of the same degree. We also conjecture that an
ideal generated by $m$'th powers of forms of degree $d$ gives the same Hilbert
series as an ideal generated by generic forms of degree $md$. We verify this in
several cases. This also gives a proof of the first conjecture in some new
cases.
