The determinantal complexity of the 4*4 permanent is 15

Abstract

The best known lower bound on the determinantal complexity of the 4x4 permanent is 10. In this short note, I prove that this value is exactly 15: the 4x4 permanent is the determinant of a 15x15 matrix of affine linear forms, and cannot be that of any smaller matrix. To do so, I must establish both a lower and upper bound. To establish the upper bound, I use Grenet's construction of size 2^n-1. Hence that construction is optimal at n=4 (and n=3). This now constitutes the first knwon exact value of the determinantal complexity of the permanent at n=4. Only the value at n=3 was exactly known, prior to this note. The general quadratic bound only gives 8 for n=4. To establish the lower bound, I work from a normal form at a zero of the permanent. In the first row (and column) of the normal form, the linear forms vanish on a linear space inside the singular locus of the permanent. A hessian inequality bounds the size of the matrix in terms of that space. Then, a matrix of sie 14 foces the space into any of nine corordinate types. The forcing is either up to symmetry, or else into a family that degenerates to one of them. I then rule out every case. All involved computations are exact. In respect of the claim, I have made all certificates public.