An introduction to semialgebraic geometry by Coste M.

By Coste M.

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Then there are continuous semialgebraic functions ξ1 < . . < ξ : C → R, such that, for every x ∈ C, the set {ξ1 (x ), . . , ξ (x )} is the set of real roots of all nonzero polynomials P1 (x , Xn ), . . , Pr (x , Xn ). The graph of each ξi , and each band of the cylinder C × R bounded by these graphs, are connected semialgebraic sets, semialgebraically homeomorphic to C or C × (0, 1), respectively, and (P1 , . . , Pr )-invariant. d. of Rn−1 adapted to PROJ(P1 , . . d. d. of Rn adapted to (P1 , .

The reader who wants to learn more about model theory and its application to real algebraic geometry is invited to read the lecture notes [Pr]. Remark. One should pay attention to the fact that the quantified variables (or n-tuples of variables) have to range over R, or Rn , or possibly over a semialgebraic subset of Rn . ). 1 Semialgebraic functions Definition and first properties Let A ⊂ Rm and B ⊂ Rn be semialgebraic sets. A mapping f : A → B is said to be semialgebraic if its graph Γf = {(x, y) ∈ A × B ; y = f (x)} is a semialgebraic subset of Rm × Rn .

Replacing S with its intersection with a ball with center x and radius 1, we can assume S bounded. Then clos(S) is a compact semialgebraic set. By the triangulation theorem, there is a finite simplicial complex K and a semialgebraic homeomorphism h : |K| → clos(S), such that x = h(a) for a vertex a of K and S is the union of some open simplices of K. In particular, since x is in the closure of S and not in S, there is a simplex σ of K whose a ◦ is a vertex, and such that h(σ) ⊂ S. Taking a linear parametrization of the segment joining a to the barycenter of σ, we obtain δ : [0, 1] → σ such that ◦ δ(0) = a and δ((0, 1]) ⊂σ.

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An introduction to semialgebraic geometry by Coste M.
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