Real Algebraic and Analytic Geometry
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Submission: 2004, May 26.
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 $Q$.
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