We prove that, for n 4, there are C ∞ nonnegative functions f of n variables (and even flat ones for n 5) which are not a finite sum of squares of C 2 functions. For n = 1, where a decomposition in a sum of two squares is always possible, we investigate the possibility of writing f = g 2 . We prove that, in general, one cannot require a better regularity than g ∈ C 1 . Assuming that f vanishes at all its local minima, we prove that it is possible to get g ∈ C 2 but that one cannot require any additional regularity.

Abstract. -We consider the 17 th Hilbert Problem for global real analytic functions in a modified form that involves infinite sums of squares. Then we prove a local-global principle for a real global analytic function to be a sum of squares of global real meromorphic functions. We deduce that an affirmative solution to the 17 th Hilbert Problem for global real analytic functions implies the finiteness of the Pythagoras number of the field of global real meromorphic functions, hence that of the field of real meromorphic power series. This measures the difficulty of the 17 th Hilbert problem in the analytic case.

It is shown that Denjoy-Carleman quasi-analytic rings of germs of functions in two or more variables fail to satisfy the Weierstrass Preparation Theorem. The result is proven via a non-extension theorem.2010 Mathematics Subject Classification. Primary 26E10; 32B05; Secondary 46E25.

Abstract. -In this work we present the concept of C-semianalytic subset of a real analytic manifold and more generally of a real analytic space. C-semianalytic sets can be understood as the natural generalization to the semianalytic setting of global analytic sets introduced by Cartan (C-analytic sets for short). More precisely S is a C-semianalytic subset of a real analytic space (X, OX ) if each point of X has a neighborhood U such that S ∩U is a finite boolean combinations of global analytic equalities and strict inequalities on X. By means of paracompactness C-semianalytic sets are the locally finite unions of finite boolean combinations of global analytic equalities and strict inequalities on X.The family of C-semianalytic sets is closed under the same operations as the family of semianalytic sets: locally finite unions and intersections, complement, closure, interior, connected components, inverse images under analytic maps, sets of points of dimension k, etc. although they are defined involving only global analytic functions. In addition, we characterize subanalytic sets as the images under proper analytic maps of C-semianalytic sets.We prove also that the image of a C-semianalytic set S under a proper holomorphic map between Stein spaces is again a C-semianalytic set. The previous result allows us to understand better the structure of the set N (X) of points of non-coherence of a C-analytic subset X of a real analytic manifold M . We provide a global geometric-topological description of N (X) inspired by the corresponding local one for analytic sets due to Tancredi-Tognoli (1980), which requires complex analytic normalization. As a consequence it holds that N (X) is a C-semianalytic set of dimension ≤ dim(X) − 2.

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