Recovery from power sums

In this talk I will speak about ongoing work with Hana Melánová and Bernd Sturmfels, in which we study the problem of recovering a collection of n numbers from the evaluation of m power sums. The polynomial system corresponds to intersecting Fermat hypersurfaces, and it can be underconstrained (m < n), square (m=n), or overconstrained (m > n). Questions that we ask are for example, when is recovery possible? If it is possible, is it unique? If it is not unique, can we give an upper bound for the number of solutions? I will present some results, and many more conjectures.