A new approach to Hilbert's theorem on ternary quartics

Victoria Powers, Bruce Reznick, Claus Scheiderer, Frank Sottile

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)617-620
Number of pages4
JournalComptes Rendus Mathematique
Issue number9
StatePublished - Nov 1 2004

ASJC Scopus subject areas

  • General Mathematics


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