On a class of power ideals (Q2341529)

Let \(S=\bigoplus_{i\geq 0}S_i\) be the polynomial ring \(\mathbb{C}[x_0,\dots, x_n]\) with standard gradation, i.e., \(S_d\) is the \(\mathbb{C}\)-vector space of forms of degree \(d\). A homogeneous ideal \(I\subset S\) is called a \textit{power ideal} if it is generated by some powers \(L_1^{d_1},\dots, L_m^{d_m}\) of linear forms and \(\mathrm{span}(L_1,\dots, L_m)=S_1\). In the paper under review the authors consider a special class of power ideals, namely, for any triple \((n,k,d)\) of positive integers and with a primitive \(k^{\mathrm{th}}\) root of unity \(\xi\), they consider the homogeneous ideal \(I_{n,k,d}\) generated by the \(k^n\) powers \((x_0+\xi^{g_1}x_1+\dots +\xi^{g_n}x_n)^{(k-1)d}\), where \(0\leq g_j\leq k-1\) for all \(j=1,\dots, n\). The quotient ring is denoted by \(R_{n,k,d}=\mathbb{C}[x_0, \dots, x_n]/I_{n,k,d}\), its homogeneous component of degree \(j\) by \([R_{n,k,d}]_j\). The main goal of the authors is to determine the Hilbert series of \(R_{n,k,d}\). They introduce a \(\mathbb{Z}_k^{n+1}\)-grading on \(R_{n,k,d}\) and using it they get a minimal set of generators for the ideal \(I_{n,k,d}\). After that they focus on the case \(k=2\) and determine the Hilbert series for \(R_{n,2,d}\). One consequence is that \([R_{n,2,d}]_{2d-1}=0\), which (in the case \(k=2\)) strenghtens a result of \textit{H. Fröberg}, \textit{G. Ottaviani} and \textit{B. Shapiro} [Proc. Natl. Acad. Sci. USA 109, No. 15, 5600--5602 (2012; Zbl 1302.11077)]. The authors also consider the case \(k>2\) and conjecture the extension of their results in the case \(k=2\). By Macaulay duality the power ideals \(I_{n,k,d}\) are related to schemes of fat points with support on the \(k^n\) points \([1:\xi^{g_1}:\dots:\xi^{g_n}]\in\mathbb{P}^n\), where \(0\leq g_j\leq k-1\) for all \(j=1,\dots, n\). The authors compute Hilbert series, Betti numbers and Gröbner basis for these \(0\)-dimensional schemes. This explicitly determine the Hilbert series of the power ideal for all \(k\). The authors have checked by several computer experiments that this agrees with their conjecture for \(k>2\).