The permanent of a square matrix

Glynn, David G.

doi:10.1016/j.ejc.2010.01.010

Cited by 111 publications

(111 citation statements)

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“…The outer sum can then be restricted to x 1 = 1, x i = ±1, i > 1, and the (−1) m replaced by (−1) m−1 . One finds that this corollary is also true for a field F of any characteristic, and it is related to the polarization identity for symmetric m-tensors; see [6,8]. It gives a formula for the permanent that should in general be as fast as Ryser's formula to calculate for larger matrices.…”

Section: Theorem 53 If S Is a Multiset Of Vectors In F M Thenmentioning

confidence: 85%

An invariant for matrices and sets of points in prime characteristic

Glynn

2010

Des. Codes Cryptogr.

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There is polynomial function X q in the entries of an m × m(q − 1) matrix over a field of prime characteristic p, where q = p h is a power of p, that has very similar properties to the determinant of a square matrix. It is invariant under multiplication on the left by a non-singular matrix, and under permutations of the columns. This gives a way to extend the invariant theory of sets of points in projective spaces of prime characteristic, to make visible hidden structure. There are connections with coding theory, permanents, and additive bases of vector spaces.

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Section: Theorem 53 If S Is a Multiset Of Vectors In F M Thenmentioning

confidence: 85%

An invariant for matrices and sets of points in prime characteristic

Glynn

2010

Des. Codes Cryptogr.

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“…This is an important example of a polynomial for which no polynomial size circuit is known (even non weakly-skew). Nevertheless, there exist formulas for computing it of much smaller size than the bounds for the general case [25,7]. For instance, Ryser's formula to compute the permanent of a matrix M is…”

Section: Comparison With Quarez's Resultsmentioning

confidence: 99%

Symmetric determinantal representation of formulas and weakly skew circuits

Grenet¹,

Kaltofen²,

Koiran³

et al. 2011

Randomization, Relaxation, and Complexity in Polynomial Equation Solving

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Abstract. We deploy algebraic complexity theoretic techniques to construct symmetric determinantal representations of formulas and weakly skew circuits. Our representations produce matrices of much smaller dimensions than those given in the convex geometry literature when applied to polynomials having a concise representation (as a sum of monomials, or more generally as an arithmetic formula or a weakly skew circuit). These representations are valid in any field of characteristic different from 2. In characteristic 2 we are led to an almost complete solution to a question of Bürgisser on the VNP-completeness of the partial permanent. In particular, we show that the partial permanent cannot be VNP-complete in a finite field of characteristic 2 unless the polynomial hierarchy collapses.

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“…If F has characteristic two ( p = 2), the permanent equals the determinant. See [2] and [3] for some of the theory of permanents.…”

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confidence: 99%