Determinant boundaries and gaps in determinantal complexity

Abstract

Landsberg, Manivel, and Ressayre constructed determinant-boundary components whose defining forms, for odd n >= 3, are limits of n x n determinants but cannot themselves be represented by determinants of that size. Their border determinantal complexity (DC) is therefore smaller than their exact DC. We prove that the difference can grow linearly with n. For every n >= 10, we construct an explicit homogeneous form of degree n, concise in n^2 variables, whose orbit closure is a boundary component, whose border DC is n, and whose exact DC is at least 2n - 1. Thus n rows suffice when limits are allowed, whereas every exact representation requires at least 2n - 1 rows. The same bound holds for the unpadded odd-skew forms at every odd n >= 3. We also prove that every concise degree-n boundary form in n^2 variables has exact DC at least n + 2. We construct three families of pairwise distinct boundary components. Together with the traceless determinant, they give at least floor((n-1)/2) + floor(n/3) components for every n >= 3, with an additional term counted explicitly in the paper. The main lower bound follows from a general degree lemma. If an affine matrix represents an irreducible homogeneous polynomial f of degree d and has globally free rank-one cokernel over the coordinate ring of f = 0, its size is at least 2d - 1. The adjugate factors into two nonzero kernel vectors, each requiring a coordinate of degree at least d - 1. Comparing their product with the degree of a cofactor proves the bound. The lemma is sharp for individual representations, as witnessed by algebraic branching programs. Whether the exact-versus-border gap can grow superlinearly remains open.