The Pierce–Birkhoff conjecture in its abstract version

Let $R$ be a real closed field, let $A=R[x_1,\dots,x_n]$, and let $\operatorname{Sper} A$ be the real spectrum of $A$. For a piecewise-polynomial function $f$ and a point $\alpha\in\operatorname{Sper} A$, choose a local polynomial representative $f_\alpha\in A$; define the separating ideal $\langle\alpha,\beta\rangle$ as the ideal generated by elements of $A$ that have opposite signs at $\alpha$ and $\beta$.

Pierce–Birkhoff conjecture, abstract version. For every piecewise-polynomial function $f$ and every $\alpha,\beta\in\operatorname{Sper} A$, if $f_\alpha$ and $f_\beta$ are local representatives of $f$ at $\alpha$ and $\beta$, respectively, then

$$f_\alpha-f_\beta\in\langle\alpha,\beta\rangle.$$

Madden showed that this local statement is equivalent to the original Pierce–Birkhoff conjecture. The source presents it as an equivalent formulation and does not report a general proof.