Let $R$ be the set of homogeneous polynomials of degree $n$ in $d$ variables over $\mathbb{C}$. When $n>2$, the set of elements of $R$ that split into a product of linear factors forms a proper subset $S$ of $R$.

- Is $S$ an algebraic variety, or something almost as nice?
- If so, how can $S$ be described implicitly, in terms of the original coefficients, without using a factorization algorithm? In other words, is there a finite set of polynomials in the coefficients which vanish if and only if $p\in S$ (or something almost as nice)?
- If so, how do I compute these polynomials for each fixed $d$ and $n$?
- Are there any other shortcuts for checking whether a homogeneous polynomial splits into linear factors?