  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.
