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On connectedness of sets in the real spectra of polynomial rings. (English) Zbl 1169.14039
The Pierce-Birkhoff conjecture is a long standing open problem in real algebra. It says that any continuous piecewise polynomial function on \(\mathbb{R}^n\) can be obtained from polynomial functions by the iterated formation of maxima and minima of functions. The question can be asked more generally for continuous piecewise polynomial functions over any real closed field \(R\), or even over any ordered field. For real closed fields the conjecture is known to be true with \(n=1\) and \(n=2\). But for \(n\geq 3\) it is open, despite serious efforts by various researchers.
Conceivably, the real spectrum provides a tool for deciding the conjecture. Madden introduced the notion of the separating ideal of two points of the real spectrum J. J. Madden [Arch. Math. 53, No. 6, 565–570 (1989; Zbl 0691.14012)]. He showed that the Pierce-Birkhoff conjecture is equivalent to a conjecture about sepearating ideals, thus putting the Pierce-Birkhoff problem in a local form. So far this approach has not led to a solution, either. The authors now introduce a connectedness conjecture. They show that the connectedness conjecture implies the Pierce-Birkhoff conjecture. The connectedness conjecture is concerned with the question whether certain subsets of the real spectrum of a polynomial ring \(R[X_1,\dots,X_n]\) are topologically connected. The authors do not decide the connectedness conjecture, but take first steps in this direction by proving special cases of the conjecture.

14P10 Semialgebraic sets and related spaces
13J30 Real algebra
Full Text: DOI
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