Quadratic Lower Bound for Permanent Vs. Determinant in any Characteristic

In Valiant’s theory of arithmetic complexity, the classes VP and VNP are analogs of P and NP. A fundamental problem concerning these classes is the Permanent and Determinant Problem: Given a field \mathbbF{\mathbb{F}} of characteristic ≠ 2, and an integer n, what is the minimum m such that the permanent of an n × n matrix X = (x_ij) can be expressed as a determinant of an m × m matrix, where the entries of the determinant matrix are affine linear functions of x_ij ’s, and the equality is in \mathbbFX]{\mathbb{F}}{\bf X}]. Mignon and Ressayre (2004) proved a quadratic lower bound m = W(n²)m = \Omega(n^{2}) for fields of characteristic 0. We extend the Mignon–Ressayre quadratic lower bound to all fields of characteristic ≠ 2.