  We deal with a generalization of a Theorem of P. Gordan and M. Noether on
hypersurfaces with vanishing (first) Hessian. We prove that for any given
$N\geq 3$, $d \geq 3$ and $2\leq k < \frac{d}{2}$ there are infinitely many
irreducible hypersurfaces $X = V(f)\subset \mathbb{P}^N$, of degree
$\operatorname{deg}(f)=d$, not cones and such that their Hessian determinant of
order $k$, $\operatorname{hess}^k_f$, vanishes identically. The vanishing of
higher Hessians is closely related with the Strong (or Weak) Lefschetz property
for standard graded Artinian Gorenstein algebra, as pointed out firstly in
\cite{Wa1} and later in \cite{MW}. As an application we construct for each pair
$(N.d) \neq (3,3),(3,4)$ infinitely many standard graded Artinian Gorenstein
algebras $A$, of codimension $N+1 \geq 4$ and with socle degree $d \geq 3$
which do not satisfy the Strong Lefschetz property, failing at an arbitrary
step $k$ with $2\leq k<\frac{d}{2}$. We also prove that for each pair $(N,d)$,
$N \geq 3$ and $d \geq 3$ except $(3,3)$, $(3,4)$, $(3,6)$ and $(4,4)$ there
are infinitely many standard graded Artinian Gorenstein algebras of codimension
$N+1$, with socle degree $d$, with unimodal Hilbert vectors which do not
satisfy the Weak Lefschetz property.
