Powers of generic ideals and the weak Lefschetz property for powers of some monomial complete intersections

Given an ideal $I=(f_1,\dots ,f_r)$ in $\mathbb{C}[x_1,\dots ,x_n]$ generated by forms of degree $d$, and an integer $k>1$, how large can the ideal $I^k$ be, i.e., how small can the Hilbert function of $\mathbb{C}[x_1,\dots ,x_n]\mathrm{/}{I}^k$ be? If $r\le n$ the smallest Hilbert function is achieved by any complete intersection, but for $r>n$, the question is in general very hard to answer. We study the problem for $r=n+1$, where the result is known for $k=1$. We also study a closely related problem, the Weak Lefschetz property, for $S\mathrm{/}{I}^k$, where $I$ is the ideal generated by the $d$’th powers of the variables.