An elementary proof of Hilbert%26apos;s theorem on ternary quartics

作者：Pfister Albrecht; Scheiderer Claus^*

来源：Journal of Algebra, 2012, 371: 1-25.

DOI：10.1016/j.jalgebra.2012.07.042

• Polynomials
• Sums of squares

In 1888, Hilbert proved that every nonnegative 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.

• 出版日期2012-12-1