On the ranks and border ranks of symmetric tensors
arXiv:0901.0487
Abstract
Motivated by questions arising in signal processing, computational complexity, and other areas, we study the ranks and border ranks of symmetric tensors using geometric methods. We provide improved lower bounds for the rank of a symmetric tensor (i.e., a homogeneous polynomial) obtained by considering the singularities of the hypersurface defined by the polynomial. We obtain normal forms for polynomials of border rank up to five, and compute or bound the ranks of several classes of polynomials, including monomials, the determinant, and the permanent.
v1: 22 pages; v2: 23 pages, numerous small improvements; v3: final version, accepted for publication in Found. Comp. Math