### Abstract

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 language | English (US) |
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Pages (from-to) | 617-620 |

Number of pages | 4 |

Journal | Comptes Rendus Mathematique |

Volume | 339 |

Issue number | 9 |

DOIs | |

State | Published - Nov 1 2004 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Powers, V., Reznick, B., Scheiderer, C., & Sottile, F. (2004). A new approach to Hilbert's theorem on ternary quartics.

*Comptes Rendus Mathematique*,*339*(9), 617-620. https://doi.org/10.1016/j.crma.2004.09.014