Apolarity, Hessian and Macaulay Polynomials

Abstract

A result by Macaulay states that an Artinian graded Gorenstein ring R of socle dimension one and socle degree δ can be realized as the apolar ring of a homogeneous polynomial g of degree δ in x ₀, ⋅, x _n . If R is the Jacobian ring of a smooth hypersurface f(x ₀, ⋅, x _n ) = 0, then δ is equal to the degree of the Hessian polynomial of f. In this article we investigate the relationship between g and the Hessian polynomial of f, and we provide a complete description for n = 1 and deg(f) ≤4 and for n = 2 and deg(f) ≤3.

Key Words:

• Apolarity
• Hessian polynomial
• Jacobian ring
• Macaulay correspondence

2010 Mathematics Subject Classification:

• Primary 14N15
• Secondary 14J70

ACKNOWLEDGMENTS

Both authors wish to warmly thank Prof. Edoardo Sernesi for having suggested this problem and for many helpful and fruitful discussions. The authors are also grateful to Prof. Ragni Piene and to the referee, whose suggestions improved the presentation of this article.

Notes

Communicated by C. Pedrini.