Apolarity for the determinant and permanent of generic matrices

Abstract

The Waring problem for forms asks how to write a homogeneous polynomial of degree n as a sum of nth powers of linear polynomials. The rank of a polynomial is the least number of terms in such an expression. In this paper we give a lower bound for the rank of the determinant and the permanent of a generic and generic symmetric matrix, the Pfaffian of a generic skew symmetric matrix, and the Hafnian of a symmetric matrix, using commutative algebra methods. Our main commutative algebra result is that the ideals of differential operators that are zero on the determinant and permanent of a generic matrix and the determinant of the generic symmetric matrix are ægenerated in degree two; while the apolar ideal of the permanent of a generic symmetric matrix is generated in degrees two and three. Using a result of K. Ranestad and F.O. Schreyer, we then obtain the lower bounds for the corresponding ranks.