Sums of squares of regular functions on rational surfaces

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.