  A Vinnikov curve is a projective plane curve which can be written in the form
det(xX+yY+zZ)=0 for X, Y and Z positive definite Hermitian n by n matrices.
Given three n-tuples of positive real numbers, alpha, beta and gamma, there
exist A, B and C in GL_n \CC with singular values alpha, beta and gamma and
ABC=1 if and only if there is a Vinnikov curve passing through the 3n points
(-1: alpha_i^2:0), (0:-1:beta_i^2) and (gamma_i^2:0:-1). Knutson and Tao proved
that another equivalent condition for such A, B and C to exist is that there is
a hive (defined within) whose boundary is (log alpha, log beta, log gamma). The
logarithms of the coefficients of F approximately form such a hive; this leads
to a new proof of Knutson and Tao's result. This paper uses no representation
theory and essentially no symplectic geometry. In their place, it uses Viro's
patchworking method and a topological description of Vinnikov curves.
