Special apolar subset: the case of star configurations

Abstract

In this paper we consider a generic degree $d$ form $F$ in $n+1$ variables. In particular, we investigate the existence of star configurations apolar to $F$, that is the existence of apolar sets of points obtained by the $n$-wise intersection of $r$ general hyperplanes of ${\mathbb{P}}^n$. We present a complete answer for all values of $(d,r,n)$ except for $(d,d+1,2)$ when we present an algorithmic approach.