5 problems

Let $A$ be a regular ring, let $\alpha,\beta\in\operatorname{Sper} A$, and let $g_1,\dots,g_s$ be a finite collection of elements of $A$ not belonging to…

Let $\Sigma$ be a ring and let $\alpha,\beta\in\operatorname{Sper}\Sigma$. The connectedness property at $\alpha$ and $\beta$ requires that, for every finite collection…

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…

Strong Connectedness conjecture. A regular ring $\Sigma$ has the Strong Connectedness Property at every pair of points $\alpha,\beta\in\operatorname{Sper}\Sigma$ having a common sp…

Let $A$ be a regular ring, let $\alpha,\beta\in\operatorname{Sper} A$, and assume that either $A$ is complete or $\sqrt{\langle\alpha,\beta\rangle}=\mathfrak m$. Let…