# Convex Ternary Quartics Are SOS-Convex

arXiv: https://arxiv.org/abs/2404.14440 · HTML: https://mlsanity.com/paper/2404.14440/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Apr 2024.

## Built on by (most cited first, 1 of 1)

- [SoS1: O1 and R1-Like Reasoning LLMs are Sum-of-Square Solvers](https://mlsanity.com/paper/2502.20545/) (2025)

## Abstract (arXiv)

We prove that convex ternary quartic forms are sum-of-squares-convex (sos-convex). This result is in a meaningful sense the ``convex analogue'' a celebrated theorem of Hilbert from 1888, where he proves that nonnegative ternary quartic forms are sums of squares. We show by an appropriate construction that exploiting the structure of the Hessian matrix is crucial in any possible proof of our result.