# Sums of three squares and Noether-Lefschetz loci

## TL;DR

This paper demonstrates the density of certain algebraic sets related to sums of squares and real surfaces in algebraic geometry, revealing new insights into their approximation properties and field levels.

## Contribution

It establishes the density of sums of three squares of rational functions among positive semidefinite polynomials and of certain real surfaces with function field level 2.

## Key findings

- Sums of three squares are dense among positive semidefinite polynomials in two variables.
- Real surfaces with function field level 2 are dense among those with no real points.
- New connections between sums of squares and the geometry of real algebraic surfaces.

## Abstract

We show that the set of real polynomials in two variables that are sums of three squares of rational functions is dense in the set of those that are positive semidefinite. We also prove that the set of real surfaces in P^3 whose function field has level 2 is dense in the set of those that have no real points.