2 problems

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The Separation conjecture for real spectra
Separation conjecture. If $g\text{$ in $\hspace{-.8em}/}\langle\alpha,\beta\rangle$ , then $\alpha$ and $\beta$ lie in the same connected component of
Conjectureopenconnectedness real-algebraic-geometry real-spectraarXiv posted 2 days ago
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The Pierce–Birkhoff conjecture in its abstract version
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 re…
Conjectureopenreal-algebraic-geometry real-spectraarXiv posted 2 days ago

The Separation conjecture for real spectra