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

Given an ideal I=(f1,…,fr) in C[x1,…,xn] generated by forms of degree d, and an integer k>1, how large can the ideal Ik be, i.e., how small can the Hilbert function of C[x1,…,xn]/Ik be? If r≤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/Ik, where I is the ideal generated by the d'th powers of the variables.