| 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 th 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 = &ceill0;&parl0;n+dd&parr0; /(dn + 1) &ceilr0; . It is conjectured that the secant variety is always nondefective unless d = 2 and 2 ≤ s ≤ n2 . We prove several special cases of this conjecture. In particular, we define functions s1 and s 2 such that the secant variety is nondefective when n ≥ 3 and s ≤ s1(d) or when n = 3 and s ≥ s 2(d) and a function c such that the secant variety is nondefective when d ≥ n ≥ 4 and s ≤ 2n--3 c(n, d). We further show that the secant variety is nondefective when s ≤ 30 unless d = 2 and 2 ≤ s ≤ n2 . |
