An elementary proof of Hilbert's theorem on ternary quartics

TL;DR

This paper provides an elementary proof of Hilbert's theorem on non-negative quartic forms as sums of three squares, using only basic techniques, and offers new insights into the number of such representations.

Contribution

It introduces a new elementary proof of Hilbert's theorem on ternary quartics, avoiding advanced topology and algebraic geometry methods.

Findings

Number of representations is 8 for generic forms

Number of representations is 4 for forms with a real zero

Provides elementary approach to known facts

Abstract

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.