On Strong Lefschetz Property of 0-dimensional Complete Intersections and Associated Forms

We prove that a homogeneous 0-dimensional complete intersection satisfies the Strong Lefschetz Property (SLP) in degree 1 if and only if its associated form has nonzero Hessian. This result is essentially known in the literature, but our proof differs from previous ones. We then investigate properties of the associated forms and show that they can always be reconstructed from their partial derivatives, in particular from their Jacobian ideals. As an application of our method, we prove that every smooth homogeneous polynomial which is not of Sebastiani-Thom type is uniquely determined by any nontrivial graded component of its Jacobian ideal.