An elementary proof of Hilbert's theorem on ternary quartics: Some complements

TL;DR

This paper provides a simplified, elementary proof of Hilbert's theorem that every nonnegative ternary quartic can be expressed as a sum of three squares, improving understanding and accessibility of this classical result.

Contribution

It offers further simplifications to an existing elementary proof of Hilbert's theorem on nonnegative ternary quartics.

Findings

Simplified proof using only elementary techniques

Enhanced clarity of Hilbert's theorem proof

Broader accessibility for mathematical education

Abstract

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. In a 2012 paper we presented a new approach that used only elementary techniques. In this note we add some further simplifications to this proof.