Maximal permanents, the singular locus of the permanent, and its strength

Abstract

We prove that for every k ≥ 1 the k+1 maximal permanents of a generic k×(k+1) matrix generate a radical complete intersection of codimension k+1. This establishes, in a stronger form, a conjecture of Boralevi, Carlini, Michałek and Ventura. It also shows that the ideal of maximal permanents of a generic k×N matrix has codimension N for all N > k, and that for N ≥ k+2 its components of that codimension are exactly the k zero-row subspaces. We then prove that the singular locus of the permanental hypersurface {per_n = 0} ⊂ ℂ^{n×n} has codimension exactly 2n for every n ≥ 2, and that the gradient ideal of per_n is reduced at the generic point of every component of that codimension. This answers the question of the codimension, which von zur Gathen bounded between 5 and 2n in 1987, and confirms the value predicted by Boralevi, Carlini, Michałek and Ventura. As consequences, the strength of per_n is exactly n, and for n ≥ 6 the permanent is not a sum of n products of forms whose degrees lie in [⌈n/3⌉, n − ⌈n/3⌉].