Apolarity for determinants and permanents of generic symmetric matrices

Abstract

We show that the apolar ideal to the determinant of a generic symmetric matrix is generated in degree two, and the apolar ideal to the permanent of a generic symmetric matrix is generated in degrees two and three. In each case we specify the generators of the apolar ideal. As a consequence, using a result of K. Ranestad and F. O. Schreyer we give lower bounds to the cactus rank and rank of each of these polynomials. We compare these bounds with those obtained by J. Landsberg and Z. Teitler.