On a class of power ideals

In this paper we study the class of power ideals generated by the knkn forms (x0+ξg1x1+…+ξgnxn)(k−1)d(x0+ξg1x1+…+ξgnxn)(k−1)d where ξ is a fixed primitive k  th-root of unity and 0≤gj≤k−10≤gj≤k−1 for all j  . For k=2k=2, by using a Zkn+1-grading on C[x0,…,xn]C[x0,…,xn], we compute the Hilbert series of the associated quotient rings via a simple numerical algorithm. We also conjecture the extension for k>2k>2. Via Macaulay duality, those power ideals are related to schemes of fat points with support on the knkn points [1:ξg1:…:ξgn][1:ξg1:…:ξgn] in PnPn. We compute Hilbert series, Betti numbers and Gröbner basis for these 0-dimensional schemes. This explicitly determines the Hilbert series of the power ideal for all k  : that this agrees with our conjecture for k>2k>2 is supported by several computer experiments.