An algebraic construction of the ring of piecewise polynomial functions (Q1279847)

\textit{N. Schwartz} [in: Ordered algebraic structures, The 1991 Conrad Conf., 169-202 (1993; Zbl 0827.13008)] gave the following definitions: Let \(A\) be a ring or \(\mathbb{Q}\)-algebra contained in its real closure \(R(A)\), the so-called ring of abstract semialgebraic functions. For \(\alpha\) in the real spectrum \(\text{Sper}(A)\), we denote by \(\rho(\alpha)\) the real closure of the quotient field \(qf(A/\text{supp}(\alpha))\) with respect to the total order specified by \(\alpha\). This makes \(\prod \rho(\alpha)\) a lattice ordered ring with respect to the pointwise lattice order containing \(R(A)\). An \(a \in R(A)\) is piecewise polynomial if the real spectrum \(\text{Sper}(A)\) of \(A\) can be covered by finitely many closed constructible sets \(C_1, \ldots, C_r\) such that for certain \(a_1, \ldots, a_r\in A\) there holds \(a |C_i = a_i\) for all \(i\). Denoting the collection of piecewise polynomial functions by \(PW(A)\) and the collection of elements generated in the sense below by the elements of \(A\), the so-called sup-inf definable functions, by \(L(A)\), one has a chain \(L(A)\subseteq PW(A) \subseteq R(A) \subseteq \prod \rho (\alpha)\) of lattice ordered (sub)rings. A ring \(A\) is Pierce-Birkhoff if \(L(A) = PW(A)\). Defining inductively \(L_0(A) = A\), \(L_{n+1}(A) = L_n(A) [|a |: a \in L_n(A)]\), one has \(L(A) = \bigcup_n L_n (A)\). The unresolved conjecture these authors made in 1956 [cf. \textit{G. Birkhoff} and \textit{R. S. Pierce}, Anais Acad. Bras. Cic. 28, 41-69 (1956; Zbl 0070.26602)] is that this equality holds for the case \(A=\mathbb{Q}[X_1,\dots,X_n]\), in which indeed \(L(A)\) and \(PW(A)\) assume the meaning the terminology suggests [see \textit{J. J. Madden}, Arch. Math. 53, No. 6, 565-570 (1989; Zbl 0691.14012) or \textit{C. N. Delzell}, Rocky Mt. J. Math. 53, No. 3, 651-668 (1989; Zbl 0715.14047)]. In the paper under review a construction corresponding to the inductive algebraic construction of \(L(A)\) is provided for \(PW(A)\). Interesting corollaries for understanding the Pierce-Birkhoff conjecture are derived. Assume (real) intermediate rings \(A \subset B\subset C\subset PW(A)\). Let \((\text{Spec}B)_{\text{re}}=\{\text{real primes of }B\)