Expressing Polynomials as the Permanent of low rank Square Matrices

It is known that the problem of computing the permanent of a given matrix is #P hard. However, Alexander I. Barvinok has proven that if we fix the rank of the matrix then its permanent can be computed in strongly polynomial time. Barvinokʹs algorithm [Barvinok, A. I., Two algorithmic results for the traveling salesman problem, Math. Oper. Res. 21 (1996), 65–84.] computes the permanent of square matrices of fixed rank by constructing polynomials. We study the problem of expressing polynomials as the permanent of low rank square matrices and vice versa. We prove that the permanent of a square matrix with rank 1 is a monomial and the permanent of a square matrix (with integer entries) that has not full rank, is a polynomial with even coefficients. We also prove that, for a polynomial f ∈ k [ x ] , there exist a square matrix of rank 2, whose permanent is the polynomial f. Our results contribute in computing the permanent of a square matrix efficiently.