On polynomials with given Hilbert function and applications
  Using Macaulay's correspondence we study the family of Artinian Gorenstein local algebras with fixed symmetric Hilbert function decomposition. As an application we give a new lower bound for cactus varieties of the third Veronese embedding. We discuss the case of cubic surfaces, where interesting phenomena occur. 
Alessandra Bernardi, Joachim Jelisiejew, Pedro Macias Marques,   Kristian Ranestad
math.AC math.AG