# Permanent formulae from the Veronesean

@article{Glynn2013PermanentFF, title={Permanent formulae from the Veronesean}, author={David G. Glynn}, journal={Designs, Codes and Cryptography}, year={2013}, volume={68}, pages={39-47} }

The two formulae for the permanent of a d × d matrix given by Ryser (1963) and Glynn (2010) fit into a similar pattern that allows generalization because both are related to polarization identities for symmetric tensors, and to the classical theorem of P. Serret in algebraic geometry. The difference between any two formulae of this type corresponds to a set of dependent points on the “Veronese variety” (or “Veronesean”) vd([d − 1]), where vd([n]) is the image of the Veronese map vd acting on [n…

## 15 Citations

### Some facts on Permanents in Finite Characteristics

- MathematicsArXiv
- 2017

The following paper extends the polynomial-time computability of the permanent over fields of characteristic 3 for k-semi-unitary matrices to study more closely the case k > 1 regarding the (n-k)x-sub-permanents (or permanent-minors) of a unitary nxn-matrix and their possible relations.

### Parameterized Applications of Symbolic Differentiation of (Totally) Multilinear Polynomials

- Mathematics, Computer ScienceICALP
- 2021

This work gives faster parameterized algorithms for the matroid k-parity and k-matroid intersection problems for linear matroids, and faster deterministic algorithms for several problems, including the first deterministic polynomial time algorithm for testing if a linear space of matrices of logarithmic dimension contains an invertible matrix.

### Veroneseans, power subspaces and independence

- Mathematics
- 2011

Results are proved indicating that the Veronese map v_d often increases independence of both sets of points and sets of subspaces. For example, any d+1 Veronesean points of degree d are independent.…

### A Faster Hafnian Formula for Complex Matrices and Its Benchmarking on a Supercomputer

- Computer ScienceACM J. Exp. Algorithmics
- 2019

This work introduces new and simple algorithms for the calculation of the number of perfect matchings of complex weighted, undirected graphs with and without loops that run in O(n3 2n/2) time, and are the fastest exact algorithms to compute these quantities.

### Computing matrix permanent with collective boson operators

- Computer Science
- 2016

A generalized algorithm for computing permanents is developed that can handle the arbitrary matrices with repeated columns and rows and can be expressed as a sampling problem like Gurvits's randomized algorithm.

### An Algorithmic Method of Partial Derivatives

- Mathematics, Computer ScienceArXiv
- 2020

A new complexity measure is studied on the space of homogeneous polynomials, namely the bilinear complexity of a polynomial's apolar algebra, which is based on the Waring rank and the exponent of matrix multiplication.

### A faster hafnian formula for complex matrices and its benchmarking on the Titan supercomputer

- Computer ScienceArXiv
- 2018

This work introduces new and simple algorithms for the calculation of the number of perfect matchings of complex weighted, undirected graphs with and without loops, which are the fastest exact algorithms to compute these quantities.

### Generalized concurrence in linear optical computing: complexity analysis of a matrix permanent algorithm

- Computer Science
- 2016

A generalized algorithm for computing permanents that can handle arbitrary matrices with repeated columns and rows, which can appear in the simulation of Fock state boson sampling with an arbitrary number of photons per mode is proposed.

### Waring Rank, Parameterized and Exact Algorithms

- Computer Science, Mathematics2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)
- 2019

We show that the Waring rank (symmetric tensor rank) of a certain family of polynomials has intimate connections to the areas of parameterized and exact algorithms, generalizing some well-known…

### Approximating the Permanent by Sampling from Adaptive Partitions

- Computer ScienceNeurIPS
- 2019

ADAPART uses an adaptive, iterative partitioning strategy over permutations to convert any upper bounding method for the permanent into one that satisfies a desirable `nesting' property over the partition used, and provides significant speedups over prior work.

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