A New Proof of Hilbert's Theorem on Ternary Quartics
  David Hilbert proved that a non-negative real quartic form f(x,y,z) is the sum of three squares of quadratic forms. We give a new proof which shows that if the complex plane curve Q defined by f is smooth, then f has exactly 8 such representations, up to equivalence. They correspond to those real 2-torsion points of the Jacobian of Q which are not represented by a conjugation-invariant divisor on Q. 
Victoria Powers, Bruce Reznick, Claus Scheiderer, and Frank Sottile