# A Linear-algebraic Proof of Hilbert's Ternary Quartic Theorem

## TL;DR

This paper presents a linear-algebraic proof of Hilbert's ternary quartic theorem, demonstrating that nonnegative degree 4 homogeneous polynomials in three variables can be expressed as sums of three squares, using cone generation by rank 1 matrices.

## Contribution

It introduces a novel linear-algebraic approach to Hilbert's theorem by showing the cone of positive semidefinite matrices is generated by rank 1 elements, providing a new proof technique.

## Key findings

- Structured cone of positive semidefinite matrices is generated by rank 1 elements.
- Provides a linear-algebraic proof of Hilbert's ternary quartic theorem.
- Shows every nonnegative degree 4 homogeneous polynomial in three variables is a sum of three squares.

## Abstract

Hilbert's ternary quartic theorem states that every nonnegative degree 4 homogeneous polynomial in three variables can be written as a sum of three squares of homogeneous quadratic polynomials. We give a linear-algebraic approach to Hilbert's theorem by showing that a structured cone of positive semidefinite matrices is generated by rank 1 elements.