  T. Harima and J. Watanabe studied the Lefschetz properties of free extension
Artinian algebras $C$ over a base $A$ with fibre $B$. The free extensions are
deformations of the usual tensor product, when $C$ is also Gorenstein, so are
$A$ and $B$, and it is natural to ask for the relation among the Macaulay dual
generators for the algebras. Writing a dual generator $F$ for $C$ as a
homogeneous "polynomial" in $T$ and the dual variables for $B$, and given the
dual generator for $B$, we give sufficient conditions on $F$ that ensure that
$C$ is a free extension of $A={\sf k}[t]/(t^n)$ with fiber $B$. We give
examples that explore the sharpness of the statements. We also consider a
special set of coinvariant algebras $C$ which are free extensions of $A$, but
which do not satisfy the sufficient conditions of our main result.
