Sums of even powers of k-regulous functions

Abstract

We provide an example of a nonnegative $k$-regulous function on $R^n$ 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 $p_{2d}\left(R^k\left(R^n\right)\right)$ of $k$-regulous functions on $R^n$ for $k\ge 1$ and $n\ge 2$. We also prove that the second Pythagoras number of the ring of 0-regulous functions $R^0\left(X\right)$ on an irreducible 0-regulous affine variety $X$ is finite and bounded from above by $2^{dimX}$.