On a class of power ideals

Abstract

In this paper we study the class of power ideals generated by the $k^n$ forms $(x_0+{\xi }^{g_1}{x}_1+\dots +{\xi }^{g_n}{x}_n{)}^{(k-1)d}$ where $\xi$ is a fixed primitive $k^{th}$-root of unity and $0\le g_j\le k-1$ for all $j$. For $k=2$, by using a ${\mathbb{Z}}_k^{n+1}$-grading on $\mathbb{C}[x_0,\dots ,x_n]$, we compute the Hilbert series of the associated quotient rings via a simple numerical algorithm. We also conjecture the extension for $k>2$. Via Macaulay duality, those power ideals are related to schemes of fat points with support on the $k^n$ points $[1:{\xi }^{g_1}:\dots :{\xi }^{g_n}]$ in ${\mathbb{P}}^n$. We compute Hilbert series, Betti numbers and Gröbner basis for such $0$-dimensional schemes. This explicitly determines the Hilbert series of the power ideal for all $k$: that this agrees with our conjecture for $k>2$ is supported by several computer experiments.

References in corpus (2)

• On the Waring problem for polynomial rings
• Containment results for ideals of various configurations of points in P^N

Cited by in corpus (1)

• The initial ideal of generic sequences and Fröberg's Conjecture