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

TL;DR

This paper provides a geometric interpretation of graded Artinian Gorenstein algebras, characterizes the standard locus, and proves the Full Perazzo Conjecture by analyzing CW-complexes and Hilbert functions.

Contribution

It introduces a geometric framework for understanding Artinian Gorenstein algebras and proves the Full Perazzo Conjecture using topological and algebraic methods.

Findings

The standard locus is characterized as a subset of projective space.

The locus of full Perazzo polynomials forms a union of minimal irreducible components.

The Hilbert function attains minimal values on irreducible components of the standard locus.

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 a 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.