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Xn−1 + an−1 Xn , Xn ) such that all polynomials P1 (v(X)), . . , P (v(X)) are quasi-monic with respect to Xn . Indeed, if Pi (X) = Πi (X) + · · ·, where Πi is the homogeneous part of highest degree (say di ) of Pi , then Pi (v(X)) = Xndi Πi (a1 , . . , an−1 , 1)+ terms of lower degree with respect to Xn . It suffices to choose a1 , . . , an−1 such that none of the Πi (a1 , . . , an−1 , 1) is zero. Then we add to the list of polynomials P1 (v(X)),. . , P (v(X)) all their nonzero derivatives of every order with respect to Xn , say P +1 ,.

Tq , finitely many semialgebraic subsets of S. Then S can be decomposed as a disjoint finite union S = pi=1 Ci , where • every Ci is semialgebraically homeomorphic (and even diffeomorphic) to an open hypercube (0, 1)di , • the closure of Ci in S is the union of Ci and some Cj ’s, j = i, with dj < di , • every Tk is the union of some Si . Proof. We start with a list of polynomials (P1 , . . , P ) such that S and all Tk are described by boolean combinations of sign conditions on polynomials of this list.

Adapted to P1 , . . 3. DECOMPOSITION OF A SEMIALGEBRAIC SET 39 or real algebraic numbers. The algorithm works in the following way: • Given a list of polynomials in one variable X1 , it counts and isolates in intervals with rational endpoints all real roots of these polynomials (this can be done by using Sturm’s method). The cells of C1 are the roots and the intervals between the roots. The roots are characterized by the polynomial equation they satisfy and the interval with rational endpoints which isolates them.

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