A New Proof of Hilbert's Theorem on Ternary Quartics

Details A New Proof of Hilbert's Theorem on Ternary Quartics A New Proof of Hilbert's Theorem on Ternary Quartics

2004-05-25

arxiv.org/abs/math/0405475v1

Comptes Rendus Mathematique (Paris), 339, Issue 9, (2004), 617--620.

Mathematics

Algebraic Geometry

4 pages

Scientific paper

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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