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Regulous functions. (Fonctions régulues.) (French. English summary) Zbl 1390.14172

The manuscript proposes a new approach of the real geometry from the viewpoint of the algebraic structure of \(k\)-regulous functions (that is, rational functions admitting \(C^{k}\)-extensions). Let \(\mathcal R(\mathbb{R}^{n})\) denote the field of rational functions, that is, functions of the form \(f=p/q,\) where \(p,q\in R[x_{1},\ldots x_{n}],\) with \(q\) not vanishing on a Zariski open (therefore, dense) subset \(U\subset \mathbb{R}^{n}\). Motivated by W. Kucharz [Adv. Geom. 9, No. 4, 517–539 (2009; Zbl 1173.14341)] the authors consider for each value \(k\in \mathbb{N}\cup \{\infty \}\) the ring \(\mathcal{R}^{k}(\mathbb{R}^{n})\) of rational functions admitting a \(C^{k}\)-extension on \(\mathbb{R}^{n}\) (subring of \(\mathbb{R}(\mathbb{R}^{n})\)). These functions are called regulous for \(k=0\) (continuous-rational, according to Kucharz) and respectively, \(k\)-regulous for \(k\geq 1\). Notice that the rational function \(1/x\) is not regulous, while the rational function \[ f_{k}(x,y)=\frac{x^{3+k}}{x^{2}+y^{2}} \] is \(k\)-regulous but not \((k+1)\)-regulous. Notice further that a regulous function (and a fortiori a \(k\)-regulous function) is semi-algebraic (its graph is the Euclidean closure in \(\mathbb{R}^{n+1}\) of the graph of a rational function), and consequently, every \(\infty \)-regulous function is Nash. Based on this, the authors establish (Théorème 3.3) that the subring \(\mathcal{R}^{\infty }(\mathbb{R}^{n})\) coincides with the ring \(\mathcal{Q}(\mathbb{R}^{n})\) of regular functions (that is, rational functions with \(U=\mathbb{R}^{n}\)). This ring is Noetherian, defining the Noetherian Zariski topology, but fails to satisfy Nullstellensatz. On the other hand, the authors show that for \(k\in \mathbb{N}\), although the ring \(\mathcal{R}^{k}(\mathbb{R}^{n})\) is not Noetherian (Proposition 4.16), its Zariski spectrum \(\mathrm{Spec}(\mathcal{R}^{k}(\mathbb{R}^{n}))\) is a Noetherian topological space, and the \(k\)-regulous topology in \(\mathbb{R}^{n}\) – corresponding to closed sets of the form \(Z(f)=\{x\in \mathbb{R} ^{n}:f(x)=0\}\) with \(f\in \mathcal{R}^{k}(\mathbb{R}^{n})\) – is also Noetherian (Théorème 4.3). This is a consequence of the following result: given \(f\in \mathcal{R}^{0}(\mathbb{R}^{n})\) (regulous function), \(\mathbb{R}^{n}\) can be stratified to a finite number of Zariski-locally closed sets \(S_{i}\) such that \(f|_{S_{i}}\) is regular for all \(i\) (Théorème 4.1), which shows in addition that every \(k\)-regulous closed subset of \(\mathbb{R}^{n}\) is Zariski-constructible. Based on a \(k\)-regulous version of the Lojasiewicz inequality (Lemma 5.2), the authors establish that the reverse also holds (Théorème 6.4): The \(k\)-regulous-closed subsets of \(\mathbb{R}^{n}\) coincide with the Zariski-constructible ones. Consequently, the \(k\)-regulous topology is independent of \(k\in \mathbb{N}\) (Corollaire 6.5), but it is strictly finer than the Zariski topology (corresponding to \(k=\infty \)) for \(n\geq 2\) (see Example 3.4). Let us note that an even finer Noetherian topology is defined by the arc-symmetric sets (zeros of arc-analytic functions), see [K. Kurdyka, Math. Ann. 282, No. 3, 445–462 (1988; Zbl 0686.14027)]. Last but not least, the authors show that the rings \(\mathcal{R}^{k}(\mathbb{R}^{n})\) satisfy Nullstellensatz (Théorème 5.24) and the corresponding regulous versions of Cartan’s Theorem A and B (Théorème 5.46, Théorème 5.47).
The manuscript is written in a very pleasant way and the presentation is clear and complete.

MSC:

14P05 Real algebraic sets
14P20 Nash functions and manifolds
14E05 Rational and birational maps
14F17 Vanishing theorems in algebraic geometry
26C15 Real rational functions
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