Centre Emile Borel


September 12th – December 16th, 2005

Abstracts of the courses

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Here basic open (resp. basic closed) sets means sets of the form $ \{h_1 > 0, \ldots, h_r >0\}$ (resp.$ \{h_1 \geq 0, \ldots, h_r \geq
0\}$), where each $ h_i \in \O (V)$.
We point out that almost all these problems are still open in the general case, for dimension $ \geq 3$.
We will start by showing the relations between the solution of the $ 17^{th}$ Hilbert problem and the set of orderings $ \Sigma$ of the field of fractions or of the ring of functions $ \O$. The crucial point here is the so called Artin-Lang Property which relates, roughly speaking, the sets which are definable by $ \O$, that is, the global semianalytic sets, with the constructible sets of orderings of $ \Sigma$.

We will see that this property holds always for dimension $ \leq 2$ and for a compact manifold. As an application, we will show how to solve Hilbert problem and to prove that the closure and the connected components of a global semianalytic set in a smooth analytic surface are still global semianalytic sets.
The Artin-Lang Property is not known for non compact manifolds of dimension $ \geq 3$. However, one can solve, for instance, the finiteness problem using more traditional tools, as sheaf theory, Cartan's Theorem B, Whitney's approximation theorem. For the sake of the audience, we will recall briefly these classic technics.
At the end of the course,we will present some (partial) recent results related to the $ 17^{th}$ Hilbert Problem, the Nullstellensatz and the Positivestellensatz, some of which are still work in progress.

  1. J. BOCHNAK, M. COSTE, M.-F. ROY : Géométrie algébrique réelle, Springer (1987)
  2. C. ANDRADAS, L. BRÔCKER, J. RUIZ, Constructible sets in real geometry, Springer (1996)
  3. T.-Y. LAM, The algebraic theory of quadratic forms, Reading, Benjamin (1973)
  4. W. SCHARLAU, Quadratic and hermitian forms, Springer (1985)

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