Real Algebraic and Analytic Geometry

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106. Victoria Powers, Bruce Reznick, Claus Scheiderer, Frank Sottile:
A New Proof of Hilbert's Theorem on Ternary Quartics.

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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 $Q$.

Full text, 4p.: dvi 23k, ps.gz 138k, pdf 202k.

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