CW-complexes and minimal Hilbert vector of graded Artinian Gorenstein algebras

Abstract: I introduce a geometric interpretation of the set of standard graded Artinian Gorenstein algebras of codimension $n$ and degree $d$: the standard locus, which is a subset of the projective space of degree $d$ polynomials in $n$ variables, and I characterize it. Under opportune hypothesis, I prove that the locus of full Perazzo polynomials is the union of the minimal dimensional irreducible components of the standard locus and it is pure dimensional subset. On the other hand, I associate to any homogeneous polynomial a topological space, which is a CW-complex. Using all these sets, I prove that the Hilbert function restricted to the standard locus has minimal values on any irreducible component of the domain. I apply all this to the Full Perazzo Conjecture and I prove it.