TY - JOUR
T1 - A new approach to Hilbert's theorem on ternary quartics
AU - Powers, Victoria
AU - Reznick, Bruce
AU - Scheiderer, Claus
AU - Sottile, Frank
N1 - Funding Information:
E-mail addresses: [email protected] (V. Powers), [email protected] (B. Reznick), [email protected] (C. Scheiderer), [email protected] (F. Sottile). 1 Supported by the USAF under DARPA/AFOSR MURI Award F49620-02-1-0325. 2 Supported by European RTN-Network HPRN-CT-2001-00271 (RAAG). 3 Supported by the Clay Mathematical Institute, NSF CAREER grant DMS-0134860, and the MSRI.
PY - 2004/11/1
Y1 - 2004/11/1
N2 - 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.
AB - 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.
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U2 - 10.1016/j.crma.2004.09.014
DO - 10.1016/j.crma.2004.09.014
M3 - Article
AN - SCOPUS:26844496857
SN - 1631-073X
VL - 339
SP - 617
EP - 620
JO - Comptes Rendus Mathematique
JF - Comptes Rendus Mathematique
IS - 9
ER -