  We study algebraic complexity classes and their complete polynomials under
\emph{homogeneous linear} projections, not just under the usual affine linear
projections that were originally introduced by Valiant in 1979. These
reductions are weaker yet more natural from a geometric complexity theory (GCT)
standpoint, because the corresponding orbit closure formulations do not require
the padding of polynomials. We give the \emph{first} complete polynomials for
VF, the class of sequences of polynomials that admit small algebraic formulas,
under homogeneous linear projections: The sum of the entries of the
non-commutative elementary symmetric polynomial in 3 by 3 matrices of
homogeneous linear forms.
  Even simpler variants of the elementary symmetric polynomial are hard for the
topological closure of a large subclass of VF: the sum of the entries of the
non-commutative elementary symmetric polynomial in 2 by 2 matrices of
homogeneous linear forms, and homogeneous variants of the continuant polynomial
(Bringmann, Ikenmeyer, Zuiddam, JACM '18). This requires a careful study of
circuits with arity-3 product gates.
