  The Chow rank of a form is the length of its smallest decomposition into a
sum of products of linear forms. For a generic form, this corresponds to
finding the smallest secant variety of the Chow variety which fills the ambient
space. We determine the Chow rank of generic cubics and quaternary forms by
proving nondefectivity of all involved secant varieties. The main new
ingredient in our proof is the generalization of a technique by [Brambilla and
Ottaviani, On the Alexander--Hirschowitz theorem, J. Pure Appl. Algebra, 2008]
that consists of employing Terracini's lemma and Newton's backward difference
formula to compute the dimensions of secant varieties of arbitrary projective
varieties. Via this inductive construction, the proof of nondefectivity
ultimately reduces to proving a number of base cases. These are settled via a
computer-assisted proof because of the large dimensions of the spaces involved.
The largest base case required in our proof consisted of computing the
dimension of a vector space constructed from the $400$th secant variety of a
degree-$82$ Chow variety embedded in $\mathbb{P}^{98769}$.
