  We prove that r independent homogeneous polynomials of the same degree d
become dependent when restricted to any hyperplane if and only if their inverse
system parameterizes a variety whose (d-1)-osculating spaces have dimension
smaller than expected. This gives an equivalence between an algebraic notion
(called Weak Lefschetz Property) and a differential geometric notion,
concerning varieties which satisfy certain Laplace equations. In the toric
case, some relevant examples are classified and as byproduct we provide
counterexamples to Ilardi's conjecture.
