  Given the space $V={\mathbb P}^{\binom{d+n-1}{n-1}-1}$ of forms of degree $d$
in $n$ variables, and given an integer $\ell>1$ and a partition $\lambda$ of
$d=d_1+\cdots+d_r$, it is in general an open problem to obtain the dimensions
of the $\ell$-secant varieties $\sigma_\ell ({\mathbb X}_{n-1,\lambda})$ for
the subvariety ${\mathbb X}_{n-1,\lambda} \subset V$ of hypersurfaces whose
defining forms have a factorization into forms of degrees $d_1,\ldots,d_r$.
Modifying a method from intersection theory, we relate this problem to the
study of the Weak Lefschetz Property for a class of graded algebras, based on
which we give a conjectural formula for the dimension of $\sigma_\ell({\mathbb
X}_{n-1,\lambda})$ for any choice of parameters $n,\ell$ and $\lambda$. This
conjecture gives a unifying framework subsuming all known results. Moreover, we
unconditionally prove the formula in many cases, considerably extending
previous results, as a consequence of which we verify many special cases of
previously posed conjectures for dimensions of secant varieties of Segre
varieties. In the special case of a partition with two parts (i.e., $r=2$), we
also relate this problem to a conjecture by Fr\"oberg on the Hilbert function
of an ideal generated by general forms.
