2004/10/18 by Victoria Powers, Bruce Reznick, Claus Scheiderer +1 · 3 citations
Mathematics · #Algebraic Geometry and Number Theory #Advanced Differential Equations and Dynamical Systems #Analytic Number Theory Research
paper · doi:10.1016/j.crma.2004.09.014
openalex publication_date 2004/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/23
Hilbert proved that a non-negative real quartic form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mtext>,</mml:mtext> <mml:mi>y</mml:mi> <mml:mtext>,</mml:mtext> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> 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 .