On polynomials with given Hilbert function and applications

TL;DR

This paper investigates Artinian Gorenstein local algebras with specific Hilbert functions using Macaulay's correspondence and applies findings to bounds on cactus varieties and cubic surface phenomena.

Contribution

It introduces a new lower bound for cactus varieties of the third Veronese embedding and explores unique phenomena in cubic surfaces.

Findings

Established a new lower bound for cactus varieties

Analyzed the structure of Artinian Gorenstein algebras with fixed Hilbert functions

Identified interesting phenomena in the case of cubic surfaces

Abstract

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.