Nondefective secant varieties of varieties of completely decomposable forms

A variation of Waring’s problem from classical number theory is the question, ``What is the smallest number $s$ such that any generic homogeneous polynomial of degree $d$ in $n+1$ variables may be written as the sum of at most $s$ products of linear forms?‘’ This question may be answered geometrically by determining the smallest $s$ such that the $s$\nth secant variety of the variety of completely decomposable forms fills the ambient space. If this secant variety has the expected dimension, it is called nondefective, and $s=\lceil \left(\genfrac{}{}{0px}{}{n+d}{d}\right)\mathrm{/}(dn+1)\rceil$. It is conjectured that the secant variety is always nondefective unless $d=2$ and $2\leq s\leq\frac{n}{2}$. We prove several special cases of this conjecture. In particular, we define functions $s_1$ and $s_2$ such that the secant variety is nondefective when $n\ge 3$ and $s\le s_1(d)$ or when $n=3$ and $s\ge s_2(d)$ and a function $c$ such that the secant variety is nondefective when $d\geq n\geq 4$and$s\leq 2^{n-3}c(n,d)$. We further show that the secant variety is nondefective when $s\le 30$ unless $d=2$ and $2\le s\le \frac{n}{2}$.