An elementary proof of Hilbert's theorem on ternary quartics

Details An elementary proof of Hilbert's theorem on ternary quartics An elementary proof of Hilbert's theorem on ternary quartics

2010-09-16

arxiv.org/abs/1009.3144v1

Mathematics

Algebraic Geometry

26 pages

Scientific paper

In 1888, Hilbert proved that every non-negative quartic form f=f(x,y,z) with real coefficients is a sum of three squares of quadratic forms. His proof was ahead of its time and used advanced methods from topology and algebraic geometry. Up to now, no elementary proof is known. Here we present a completely new approach. Although our proof is not easy, it uses only elementary techniques. As a by-product, it gives information on the number of representations f=p_1^2+p_2^2+p_3^2 of f up to orthogonal equivalence. We show that this number is 8 for generically chosen f, and that it is 4 when f is chosen generically with a real zero. Although these facts were known, there was no elementary approach to them so far.

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