A new approach to Hilbert's theorem on ternary quartics

Abstract

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 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. To cite this article: V. Powers et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).