The article under review deals with an important problem in semialgebraic geometry. Let us start with the fundamental definitions in this setting. A subset \(M \subset \mathbb{R}^m\) is basic semialgebraic if it can be expressed as \(\{x \in \mathbb{R}^m, f(x)=0, g_1(x)>0, \dots, g_{\ell}(x)>0\}\), where \(f, g_1, \dots, g_{\ell}\) are polynomials of \(\mathbb{R}[x_1, \dots, x_m]\). Then, a semialgebraic set is a finite union of basic semialgebraic sets. Given two semialgebraic sets \(M \subset \mathbb{R}^m\) and \(N \subset \mathbb{R}^n\), a continuous map \(f : M \rightarrow N\) is semialgebraic if its graph is a semialgebraic subset of \(\mathbb{R}^{m+n}\). Now, let \(\mathcal{S}(M)\) be the set of semialgebraic functions on \(M\). By means of the sum and product of functions, \(\mathcal{S}(M)\) is endowed with a structure of commutative \(\mathbb{R}\)-algebra with unit. Calling \(\mathcal{S}^*(M)\) the subset of \(\mathcal{S}(M)\) consisting of bounded functions, \(\mathcal{S}^*(M)\) is an \(\mathbb{R}\)-subalgebra of \(\mathcal{S}(M)\). Throughout the paper, the authors study jointly \(\mathcal{S}(M)\) and \(\mathcal{S}^*(M)\), and they call either of them \(\mathcal{S}^{\diamond} (M)\). Then, \(\text{Spec}^{\diamond} (M)\) is the Zariski spectrum of \(\mathcal{S}^{\diamond} (M)\) with the Zariski topology, and \(\beta^{\diamond}(M)\) its set of closed points. Call \(\texttt{r}_M : \text{Spec}^{\diamond} (M) \rightarrow \beta^{\diamond} (M)\) the natural retraction. Then, take a semialgebraic map \(\pi : M \rightarrow N\). It has associated a homomorphism of \(\mathbb{R}\)-algebras \(\varphi_{\pi}^{\diamond} : \mathcal{S}^{\diamond} (N) \rightarrow \mathcal{S}^{\diamond} (M)\) defined by \(g \mapsto g \circ \pi\). This determines two continuous morphisms, \(\text{Spec}^{\diamond}(\pi) : \text{Spec}^{\diamond} (M) \rightarrow \text{Spec}^{\diamond} (N)\), \(\mathfrak{p} \mapsto (\varphi_{\pi}^{\diamond})^{-1}(\mathfrak{p})\), and \(\beta^{\diamond} (\pi) = \texttt{r}_N \circ \text{Spec}^{\diamond}(\pi) |_{\beta^{\diamond}(M)} : \beta^{\diamond}(M) \rightarrow \text{Spec}^{\diamond} (N) \rightarrow \beta^{\diamond}(N)\).
A long list of papers by the authors of the present article during the last ten years studies the relationship between the above maps \(\pi\) and \(\text{Spec}^{\diamond}(\pi)\). Here, they consider the case in which \(\pi : M \rightarrow N\) is a semialgebraic branched covering. Consider a map \(\pi : X \rightarrow Y\). It is a finite quasi-covering if it is separated, open, closed, surjective, and its fibers are finite. Then, the branching locus of \(\pi\) is the set \(\mathcal{B}_{\pi}\) of points in \(X\) at which \(\pi\) is not a local homeomorphism. The ramification set of \(\pi\) is \(\pi (\mathcal{B}_{\pi}) = \mathcal{R}_{\pi}\), and the regular locus of \(\pi\), \(X_{\text{reg}}\), is \(X \setminus \pi^{-1}(R_{\pi})\).
The authors state Definition 2.11, according to which \(\pi\) is a branched covering if \(X_{\text{reg}}\) is dense in \(X\) and each \(y \in Y\) admits a so-called special neighborhood, which they also define in Section 2 of the paper. This specific definition of branched covering is made in the article in order to avoid anomalous cases as shown in Example 2.26. In these conditions, the main Theorem 1.1 of the paper is the following:
Let \(\pi : M \rightarrow N\) be a semialgebraic map. Then, \(\pi\) is a branched covering if and only if \(\text{Spec}^{\diamond}(\pi)\) is a branched covering, if and only if \(\beta^{\diamond} (\pi)\) is a branched covering. In that case, \(\mathcal{B}_{\text{Spec}^{\diamond}(\pi)} = \text{Cl}_{\text{Spec}^{\diamond}(M)}(\mathcal{B}_{\pi})\), \(\mathcal{B}_{\beta^{\diamond}(\pi)} = \text{Cl}_{\beta^{\diamond}(M)}(\mathcal{B}_{\pi})\), \(\mathcal{R}_{\text{Spec}^{\diamond}(\pi)} = \text{Cl}_{\text{Spec}^{\diamond}(N)}(\mathcal{R}_{\pi})\), \(\mathcal{R}_{\beta^{\diamond}(\pi)} = \text{Cl}_{\beta^{\diamond}(N)}(\mathcal{R}_{\pi})\), where \(\text{Cl}_X(A)\) stands for the closure of \(A\) in \(X\).
The other main goal of the article is to study the collapsing set of the spectral map \(\text{Spec}^{\diamond}(\pi)\) when \(\pi\) is a \(d\)-branched covering, that is to say, a branched covering such that the fibers of the points outside \(\mathcal{R}_{\pi}\) have constant cardinality \(d\). The collapsing set \(\mathcal{C}_{\pi}\) of \(\pi\) is defined as the set of points such that the fiber \(\pi^{-1}(\pi(x))\) is a singleton. Then, the authors study \(\mathcal{C}_{\text{Spec}^{\diamond}(\pi)}\) and \(\mathcal{C}_{\beta^{\diamond}(\pi)}\). In order to do that, a map \(\mu ^{\diamond} : \mathcal{S} ^{\diamond} (M) \rightarrow \mathcal{S} ^{\diamond} (N)\) is defined in Section 4 (Definition 4.1). The result is Theorem 1.2, according to which, given a semialgebraic \(d\)-branched covering \(\pi : M \rightarrow N\), then \(\mathcal{C}_{\text{Spec}^{\diamond}(\pi)}\) is the set of prime ideals of \(\mathcal{S}^{\diamond} (M)\) containing \(\text{ker}(\mu ^{\diamond})\), and it equals \(\text{Cl}_{\text{Spec}^{\diamond}(M)}(\mathcal{C}_{\pi})\), and \(\mathcal{C}_{\beta^{\diamond}(\pi)}\) is the set of maximal ideals of \(\mathcal{S}^{\diamond} (M)\) containing \(\text{ker}(\mu ^{\diamond})\), and it equals \(\text{Cl}_{\beta^{\diamond}(M)}(\mathcal{C}_{\pi}).\)
Section 2 of the paper is devoted to define and study branched coverings, and Section 3 to analyze the properties of spectral maps associated to quasi-finite coverings and branched coverings. Then, the proof of Theorem 1.2 is developed in Section 4, and that of Theorem 1.1 in Section 5. Finally, an example is detailed in an Appendix at the end of the paper. It is worth to note that the article is very carefully written, and its introduction is very enlightening both on the state-of-the-art, and on the purpose and procedures of the work.