Sums of squares of regular functions on rational surfaces

TL;DR

This paper investigates sums of squares on cylinders over factorial curves and rational surfaces, establishing key equalities and bounds for Pythagoras numbers in algebraic geometry contexts.

Contribution

It proves the equality of Pythagoras numbers for regular functions on cylinders and rational function fields, and bounds the Pythagoras number for real rational surfaces.

Findings

Pythagoras number of regular functions on cylinders equals that of rational functions.

Bound of 12 for Pythagoras number on real nonsingular rational surfaces.

Results apply to (weakly) factorial curves and rational varieties.

Abstract

We study the sums of squares on cylinders of the form $X \times \mathbb{A}_K$ for a (weakly) factorial curve $C$. We prove the equality of the Pythagoras numbers of the ring of regular functions on the cylinder with that of the field of rational functions. We then apply these results to the case of (uniformly) rational varieties. We show that if $X$ is a nonsingular rational algebraic surface over the reals, then the Pythagoras number of the ring of regular functions on $X$ is bounded above by 12.