On points at infinity of real spectra of polynomial rings (Q841555)

Let \(R\) be a real closed field, and let \(B\) be an \(R\)-algebra. The real spectrum of \(B\), denoted by Sper\(B\), is the collection of all pairs \(\alpha = (P_{\alpha}, \leqslant_{\alpha})\), where \(P_{\alpha}\) is a prime ideal of \(B\) and \(\leqslant_{\alpha}\) is a total ordering of the quotient ring \(B/P_{\alpha}\). The ordered quotient ring \((B/P_{\alpha}, \leqslant_{\alpha})\) is denoted by \(B[\alpha]\), while \(f(\alpha)\) is the natural image of an \(f \in B\) in \(B[\alpha]\). A point \(\alpha \in\) Sper\(B\) is said to be finite if for each \(f \in B\), there is an \(N \in R\) such that \(|f(\alpha)| < N\). Otherwise, the \(\alpha\) is called a point at infinity. Let Sper\(^*B\) denote the subset of Sper\(B\) consisting of all finite points. Clearly, the polynomial ring \(R[X_1, \dots , X_n]\) is an \(R\)-algebra. In this paper under review, the authors use valuation theory to prove that Sper\(R[X_1, \dots , X_n]\) can be expressed as a finite disjoint union of subsets, each of which is homeomorphic to a subset of Sper\(^*C\) of another polynomial ring \(C\) over \(R\). For example, Sper\(R[X]\) has only two points at infinity, which can be viewed as finite points of Sper\(R[{\frac{1}{X}}]\) of the polynomial ring \(R[{\frac{1}{X}}]\).