Sums of even powers of k-regulous functions

TL;DR

This paper investigates the representation of nonnegative k-regulous functions as sums of squares, providing examples, bounds for Pythagoras numbers, and finiteness results for certain rings of functions.

Contribution

It presents the first example of a nonnegative k-regulous function not expressible as a sum of squares and establishes bounds for Pythagoras numbers of k-regulous functions.

Findings

Existence of nonnegative k-regulous functions not sum of squares

Lower bounds for Pythagoras numbers p_{2d}( ext{R}^k( ext{R}^n))

Finiteness and upper bounds for the second Pythagoras number of ext{R}^0(X)

Abstract

We provide an example of a nonnegative $k$-regulous function on $\mathbb{R}^n$ for $k\geq 1$ and $n \geq 2$ 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 $k\geq 1$ and $n\geq 2$. 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 $2^{\dim X}$.