Pierce-Birkhoff rings
The author shows an equivalent formulation of the Pierce-Birkhoff conjecture in terms of the set of separating functions of points of the real spectrum. Precisely, let A denote a commutative ring with unit and SA(A) its ring of abstract continuous semialgebraic functions. Set PW(A)\(\subset SA(A)\) for the subring of functions which are piecewise defined by elements of A. Then it is shown that the following are equivalent: (a) A verifies the Pierce-Birkhoff conjecture, that is, any \(t\in PW(A)\) is of the form \(t=\sup_{i}\{\inf_{j}(b_{ij})\}\) for some \(b_{ij}\in A.\) (b) For any \(t\in PW(A)\) and \(\gamma,\delta \in Spec_ R(A)\), there is \(h\in A\) such that h(\(\gamma)\)\(\geq t(\gamma)\) and h(\(\delta)\)\(\leq t(\delta).\) (c) For any \(t\in PW(A)\) and \(\gamma,\delta \in Spec_ R(A)\), \(t_{\gamma}-t_{\delta}\in <\gamma,\delta >\subseteq A\) is the ideal generated by all functions \(a\in A\) with a(\(\gamma)\)\(\geq 0\) and a(\(\delta)\)\(\leq 0.\) Then the author applies this characterization to show that fields and Dedekind domains satisfy the Pierce-Birkhoff conjecture, improving a result of C. N. Delzell.
separating functions of points of the real spectrum
scientific article; zbMATH DE number 4130538
scientific article