Weak normalization and seminormalization in real algebraic geometry

This version contains some inaccuracies about centrality issues. A new version will appear soon.We define the weak normalization and the seminormalization of a central real algebraic variety. The study is related to the properties of the rings of continuous rational functions and regulous functions on real algebraic varieties. We provide in particular several characterizations (algebraic or geometric) of these varieties, and study in full details the case of curves. The concept of weak normalization of a complex analytic variety has been introduced by Andreotti & Bombieri [3] in order to study the space of analytic cycles associated with a complex algebraic variety. The operation of weak normalization consists in enriching the sheaf of holomorphic functions with those continuous functions which are also meromorphic. Later Andreotti & Norguet [4] defined the notion of weak normalization in the context of schemes. For algebraic varieties, it consists roughly speaking of an intermediate algebraic variety between an algebraic variety X and its normalization, in such a way that the weak normalization of X is in bijection with X. One way to construct it is to identify in the normalization all the points belonging to the pre-image of points in X. It gives rise to a variety satisfying a universal property among those varieties in birational bijection via a universal homeomorphism onto X. The theory of seminormalization, closely related to that of weak normalization, have been developed later by Traverso [33] for commutative rings, with subsequent work notably by Swan [32] or Leahy & Vitulli [23] (see also [34]), with a more particular focus on the algebraic approach or the singularities. Note however that in the geometric context of complex algebraic variety, weak normalization and seminormalization lead to the same notion. We refer to Vitulli [35] for a survey on weak normality and seminormality for commutative rings and algebraic varieties. More recently, the concept of seminormalization is used in the minimal model program of Kollár and Kovács [16] and it appears also in [17]. In the context of real geometry, the first occurrence of weak normality or seminormality is the work by Acquistapace, Broglia and Tognoli [1] in the case of real analytic spaces. In [24] the Traverso seminormalization of real algebraic varieties is studied by considering the ring of regular functions, showing that such notion does not provide natural universal property. Seminormalization in the Nash context is introduced in [28]. Our aim in this paper is to provide appropriate definitions for weak normalization and seminormalization in real algebraic geometry, leading to natural universal properties. Contrarily to the complex setting, it will appear that the notions of weak normalization and seminormalization are distinct, the difference being witnessed by the behaviour of continuous rational functions on real algebraic varieties. The first focus on continuous rational function in real geometry is due to Kreisel [20] who proved that a positive answer to Hilbert seventeenth problem of representing a positive polynomial as a sum of squares of rational functions, can always be chosen among continuous functions. Besides, Kucharz [21] used this class of functions to approximate as algebraically as possible continuous maps betwee