  Let $F$ be a homogeneous polynomial in $S = \mathbb{C}[x_0,...,x_n]$. Our
goal is to understand a particular polynomial decomposition of $F$;
geometrically, we wish to determine when the hypersurface defined by $F$ in
$\mathbb{P}^n$ contains a star configuration. To solve this problem, we use
techniques from commutative algebra and algebraic geometry to reduce our
question to computing the rank of a matrix.
