Apolarity, Hessian and Macaulay polynomials

TL;DR

This paper explores the relationship between Macaulay's apolarity theory, Hessian polynomials, and Jacobian rings of smooth hypersurfaces, revealing new connections in algebraic geometry.

Contribution

It investigates the link between the apolar polynomial f and the Hessian of a smooth hypersurface g, extending Macaulay's classical results.

Findings

Established a connection between the apolar polynomial and the Hessian of g

Extended Macaulay's theorem to relate Jacobian rings and Hessian polynomials

Provided new insights into the structure of Gorenstein rings and hypersurfaces

Abstract

A result by Macaulay states that an Artinian graded Gorenstein ring R of socle dimension one and socle degree b can be realized as the apolar ring of a homogeneous polynomial f of degree b. If R is the Jacobian ring of a smooth hypersurface g=0, then b is just equal to the degree of the Hessian polynomial of g. In this paper we investigate the relationship between f and the Hessian polynomial of g.