# The Iris function and the matrix permanent

## TL;DR

This paper introduces the Iris function and presents two formulations for the matrix permanent, including a contour integral approach for complex matrices and a complexity bound for 0-1 matrices, advancing computational methods in matrix theory.

## Contribution

It defines the Iris function and provides novel formulations for the matrix permanent, including a complexity bound for computing permanents of 0-1 matrices.

## Key findings

- Contour integral formulation for complex matrices' permanent
- Complexity bound for 0-1 matrix permanent computation
- Introduction of the Iris function as a new analytical tool

## Abstract

This paper defines the Iris function and provides two formulations of the matrix permanent. The first formulation, valid for arbitrary complex matrices, expresses the permanent of a complex matrix as a contour integral of a second order Iris function over the unit circle around zero. The second formulation is defined for the restricted set of matrices with complex or "Gaussian" integer elements. Using the second formulation, the paper shows that the computation of the permanent of an arbitrary $n\times n$ 0-1 matrix is bounded by $o\left( n^{27} \left( \log \left( {{n^3}} \right) \right)^6 \left( \log_2 \left( {{n}} \right) \right)^2\right)$ binary operations.