Nondefective secant varieties of varieties of completely decomposable forms

Abstract

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 $s=\left\lceil\binom{n+d}{d}/(dn+1)\right\rceil$. 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 $n\geq 3$ and $s\leq s_1(d)$ or when $n=3$ and $s\geq s_2(d)$ 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 $s\leq 30$ unless $d=2$ and $2\leq s\leq\frac{n}{2}$.