[Paper Review] Apolarity for determinants and permanents of generic matrices

Masoumeh Sepideh Shafiei|arXiv (Cornell University)|Dec 3, 2012

Advanced Topics in Algebra7 references3 citations

TL;DR

This paper establishes that the apolar ideals of the determinant, permanent, Pfaffian, and Hafnian of generic matrices are all generated in degree two, providing explicit generators and Gröbner bases. Using a result by Ranestad and Schreyer, it derives lower bounds for the cactus rank and rank of these invariants, showing they are not connected sums for $ n \geq 3 $ or $ n \geq 6 $, respectively.

ABSTRACT

We show that the apolar ideals to the determinant and permanent of a generic matrix, the Pfaffian of a generic skew symmetric matrix and the Hafnian of a generic symmetric matrix are each generated in degree two. In each case we specify the generators and a Gröbner basis 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 invariants. We compare these bounds with those obtained by J. Landsberg and Z. Teitler.

Motivation & Objective

• To determine the generating degree of the apolar ideal for the determinant and permanent of a generic $ n \times n $ matrix.
• To extend this analysis to the Pfaffian of a generic skew-symmetric matrix and the Hafnian of a generic symmetric matrix.
• To compute explicit generators and Gröbner bases for the apolar ideals of these invariants.
• To apply the Ranestad–Schreyer result to derive lower bounds for cactus rank and rank of these invariants.

Proposed method

• Use Macaulay’s inverse system (apolar pairing) to define the apolar ideal $ \mathrm{Ann}(F) \subset S = \mathsf{k}[d_{ij}] $ for a homogeneous polynomial $ F \in R = \mathsf{k}[a_{ij}] $.
• Leverage the contraction action $ h \circ F = 0 $ for $ h \in S_k $, where $ h $ annihilates $ F $, and relate this to minors and permanents via known duality identities.
• Prove that $ \mathrm{Ann}(\det A)_k = M_k(A)^\perp $, where $ M_k(A) $ is the space of $ k \times k $ minors.
• Construct a Gröbner basis for the apolar ideal using $ 2 \times 2 $ subpermanents and monomials that are unacceptable under diagonal orderings.
• Apply Buchberger’s algorithm to verify that the union of subpermanents and unacceptable monomials forms a Gröbner basis.
• Use the Ranestad–Schreyer theorem to derive lower bounds on cactus rank and rank from the generating degree of the apolar ideal.

Experimental results

Research questions

• RQ1 Are the apolar ideals of the determinant and permanent of a generic $ n \times n $ matrix generated in degree two?
• RQ2 What is the structure of the Gröbner basis for the apolar ideal of the determinant and permanent of a generic matrix?
• RQ3 Do the Pfaffian of a generic skew-symmetric matrix and the Hafnian of a generic symmetric matrix also have apolar ideals generated in degree two?
• RQ4 Can the generating degree of the apolar ideal be used to bound the cactus rank and rank of these invariants?
• RQ5 Are these invariants connected sums, and how does the apolar ideal structure relate to this property?

Key findings

• The apolar ideal of the determinant of a generic $ n \times n $ matrix is generated in degree two, with generators given by $ 2 \times 2 $ subpermanents and unacceptable monomials.
• The apolar ideal of the permanent of a generic $ n \times n $ matrix is also generated in degree two, with the same set of generators.
• For the Pfaffian of a generic skew-symmetric matrix, the apolar ideal is generated in degree two, and a Gröbner basis is explicitly constructed.
• For the Hafnian of a generic symmetric matrix, the apolar ideal is generated in degree two, and a Gröbner basis is provided.
• The cactus rank and rank of the determinant, permanent, Pfaffian, and Hafnian are bounded below by the degree of the apolar ideal generators and the dimension of the apolar algebra.
• The determinant and permanent of size $ n \geq 3 $, and the Pfaffian and Hafnian of size $ n \geq 6 $, are not connected sums, as their apolar ideals are generated in degree two.