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A New Proof of Hilbert's Theorem on Ternary Quartics

Victoria Powers·Bruce Reznick·Claus Scheiderer·Frank Sottile
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Abstract

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.

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Victoria Powers, Bruce Reznick, Claus Scheiderer, Frank Sottile. 2004-05-25. A New Proof of Hilbert's Theorem on Ternary Quartics. https://arxiv.org/abs/math/0405475

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