Pierce–Birkhoff conjecture for regular rings

Let $\Sigma$ be a ring. A piecewise-polynomial function on $\operatorname{Sper}\Sigma$ is an element of $PW(\Sigma)$, and $\Sigma$ is a Pierce–Birkhoff ring if every $g\in PW(\Sigma)$ can be written as a finite supremum of finite infima of elements of $\Sigma$. Pierce–Birkhoff conjecture for regular rings. Every regular ring $\Sigma$ is a Pierce–Birkhoff ring. This is the ring-theoretic formulation of the Pierce–Birkhoff problem and is related to connectedness and separation properties of the real spectrum.