Some Remarks on Hessians and Wronskians

The purpose of this note is to elaborate on the apparent connection between Wronskians and Hessians. More generally, to a given subspace of homogeneous bivariate forms over the complex numbers, we associate two determinantal polynomials called the $W$-polynomial and the $\hat{W}$-polynomial. We give expansion and factorization formulas for these polynomials, and study their behavior under change of coordinates and duality. As an application, we give another proof of Iarrobino's theorem on the strong Lefschetz property for standard graded Artinian Gorenstein algebras in codimension two.