Sums of even powers of k-regulous functions

Abstract

We provide an example of a nonnegative $k$-regulous function on ${\mathbb{R}}^n$ for $k\ge 1$ and $n\ge 2$ which cannot be written as a sum of squares of $k$-regulous functions. We then obtain lower bounds for Pythagoras numbers $p_{2d}({\mathcal{R}}^k({\mathbb{R}}^n))$ of $k$-regulous functions on ${\mathbb{R}}^n$ for $k\ge 1$ and $n\ge 2$. We also prove that the second Pythagoras number of the ring of $0$-regulous functions ${\mathcal{R}}^0(X)$ on an irreducible $0$-regulous affine variety $X$ is finite and bounded from above by $2^{\dim X}$.

Final version, to appear in Indagationes Mathmaticae

Cited by in corpus (2)

• Sums of squares on hypersurfaces
• On Waring numbers of henselian rings