Apolarity, Hessian and Macaulay Polynomials

A result by Macaulay states that an Artinian graded Gorenstein ring R of socle dimension one and socle degree δ can be realized as the apolar ring of a homogeneous polynomial g of degree δ in x 0, ⋅, x n . If R is the Jacobian ring of a smooth hypersurface f(x 0, ⋅, x n ) = 0, then δ is equal to the degree of the Hessian polynomial of f. In this article we investigate the relationship between g and the Hessian polynomial of f, and we provide a complete description for n = 1 and deg(f) ≤4 and for n = 2 and deg(f) ≤3.