  Let ${\mathbb C}[x_1,\dots,x_n]_{d+1}$ be the vector space of homogeneous
forms of degree $d+1$ on ${\mathbb C}^n$, with $n,d\ge 2$. In earlier articles
by J. Alper, M. Eastwood and the author, we introduced a morphism, called $A$,
that assigns to every nondegenerate form the so-called associated form lying in
the space ${\mathbb C}[y_1,\dots,y_n]_{n(d-1)}$. One of the reasons for our
interest in $A$ is the conjecture---motivated by the well-known Mather-Yau
theorem on complex isolated hypersurface singularities---asserting that all
regular ${\mathrm {GL}}_n$-invariant functions on the affine open subvariety
${\mathbb C}[x_1,\dots,x_n]_{d+1,\Delta}$ of forms with nonvanishing
discriminant can be obtained as the pull-backs by means of $A$ of the rational
${\mathrm {GL}}_n$-invariant functions on ${\mathbb C}[y_1,\dots,y_n]_{n(d-1)}$
defined on ${\mathrm {im}}(A)$. The morphism $A$ factors as $A={\mathbf A}\circ
{\mathrm {grad}}$, where ${\mathrm {grad}}$ is the gradient morphism and
${\mathbf A}$ assigns to every $n$-tuple of forms of degree $d$ with
nonvanishing resultant a form in ${\mathbb C}[y_1,\dots,y_n]_{n(d-1)}$ defined
analogously to $A(f)$ for a nondegenerate $f$. In order to establish the
conjecture, it is important to study the image of ${\mathbf A}$. In the present
paper, we show that ${\mathrm {im}}({\mathbf A})$ is an open subset of an
irreducible component of each of the so-called catalecticant varieties $V$,
${\mathrm {Gor}}(T)$ and describe the closed complement to ${\mathrm
{im}}({\mathbf A})$, at the same time clarifying and extending known results on
these varieties. Furthermore, for $n=3$, $d=2$ we give a description of the
complement to ${\mathrm {im}}({\mathbf A})$ via the zero locus of the Aronhold
invariant of degree 4, which is analogous to the case $n=2$ where this
complement is known to be the vanishing locus of the catalecticant for any
$d\ge 2$.
