An introduction to semialgebraic geometry by Coste M.

By Coste M.

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The proof is completed. 60 CHAPTER 3. TRIANGULATION OF SEMIALGEBRAIC SETS Chapter 4 Families of semialgebraic sets. 1 Semialgebraic triviality of families Hardt’s theorem Let A ⊂ Rn be a semialgebraic set, defined by a boolean combination of sign conditions on polynomials P1 , . . , Pq . d. of Rn adapted to P1 , . . , Pq . The set A is a union of graphs and bands in cylinders Ci × R, where Rn−1 = C1 ∪ . . ∪ Cr is a finite semialgebraic partition. 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 .

If f is one-to-one, then dim f (S) = dim S. 3. DIMENSION 57 Proof. Let A ⊂ Rn+k be the graph of f . From the preceding lemma, it follows that dim(S) = dim(A) and dim(f (S)) ≤ dim(A), with moreover dim(f (S)) = dim(A) if f is one-to-one. 19 Let S ⊂ Rn be a semialgebraic set, x ∈ S. Show that there exist a neighborhood V of x in Rn and a nonnegative integer d such that, for every semialgebraic neighborhood W ⊂ V of x in Rn , dim(W ∩ S) = d. The integer d is called the dimension of S at x and denoted by dimx S.

Pq . d. of Rn adapted to P1 , . . , Pq . The set A is a union of graphs and bands in cylinders Ci × R, where Rn−1 = C1 ∪ . . ∪ Cr is a finite semialgebraic partition. 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.

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