AN UPPER BOUND FOR THE PERMANENT VERSUS DETERMINANT PROBLEM

The Permanent versus Determinant problem is the following: Given an n × n matrix X of indeterminates over a field of characteristic different from two, find the smallest matrix M whose coefficients are linear functions in the indeterminates such that the permanent of X equals the determinant of M. We prove that the dimensions of M are at most 2n − 1. The determinant and the permanent of an (n × n) matrix of indeterminates X = (xij)1≤i,j≤n over a field K are the two very similar polynomials defined respectively by det(X) = ∑ σ∈Sn ε(σ) n ∏ i=1 xiσ(i) and per(X) = ∑ σ∈Sn n