Apolarity for determinants and permanents of generic symmetric matrices

TL;DR

This paper investigates the algebraic structure of apolar ideals for determinants and permanents of generic symmetric matrices, providing explicit generators and deriving bounds for their rank and cactus rank.

Contribution

It establishes that the apolar ideal of the determinant is generated in degree two, while that of the permanent is generated in degrees two and three, with explicit generators identified.

Findings

Apolar ideal of the determinant is generated in degree two.

Apolar ideal of the permanent is generated in degrees two and three.

Lower bounds for rank and cactus rank are derived and compared with existing bounds.

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.