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Pn,sn (x, Xn ) as functions of x ∈ Ci . We know that S and the Sk are unions of graphs of ξi,j and bands of the cylinders Ci × R bounded by these graphs. Denote by π : Rn → Rn−1 the projection onto the space of the first n − 1 coordinates. The set π(S) is compact, semialgebraic, and it is the union of some Ci ; similarly, each π(SK ) is the union of some Ci . By the inductive assumption, there is a triangulation g : |L| → π(S), where L is a finite simplicial complex in Rn−1 and g a semialgebraic homeomorphism, such that each Ci ⊂ π(S) is the union of images by g of open simplices of L.

Xn−1 ]. One obtains Cn−1 , which is a partition of Rn−1 in PROJ(P1 , . . , Pr )-invariant cells, and a test point aC for each cell C ∈ Cn−1 . 19 to cut the cylinder C × R in (P1 , . . , Pr )-invariant cells. In order to know how many cells there are in this cylinder and to produce a test point for each cell, one computes the real roots of P1 (aC , Xn ), . . , Pr (aC , Xn ). Sturm’s method can be used once again, but the coefficients of the polynomials may be real algebraic numbers. We encounter here the problem of coding real algebraic numbers and computing with them.

Each A ∩ (Ci × R) is semialgebraically homeomorphic to a product Ci × Fi , where Fi is a semialgebraic subset of R: one can take for instance Fi = p−1 (bi ), where p : A → Rn−1 is the restriction of the projection onto the space of the n − 1 first coordinates, and bi , a point chosen in Ci . Hence, we have decomposed the target space Rn−1 as the disjoint union of finitely many semialgebraic subsets Ci , such that p is semialgebraically trivial over each Ci in the following sense. A continuous semi-algebraic mapping p : A → Rk is said to be semialgebraically trivial over a semialgebraic subset C ⊂ Rk is there is a semialgebraic set F and a semialgebraic homeomorphism h : p−1 (C) → C × F , such that the composition of h with the projection C × F → C is equal to the restriction of 61 62 CHAPTER 4.

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Encyclopaedie der mathematischen Wissenschaften und Anwendungen. Geometrie

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