By Coste M.

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If ε ∈ {−1, 0, 1} is a sign, we denote {0} if ε = 0, if ε = 1, ε = {0, 1} {0, −1} if ε = −1. 2 (Thom’s lemma) Let P1 , . . , if the derivative Pi is nonzero, there is j such that Pi = Pj ). For ε = (ε1 , . . , εs ) ∈ {−1, 0, 1}s , let Aε ⊂ R be deﬁned by Aε = {x ∈ R ; sign(Pi (x)) = εi for i = 1, . . , s} . Then • either Aε = ∅, • or Aε is a point (necessarily, at least one of the εi is 0), • or Aε is a nonempty open interval (necessarily, all εi are ±1). Let Aε = {x ∈ R ; sign(P1 (x)) ∈ ε1 and .

D. of Rn−1 adapted to PROJ(P1 , . . d. d. of Rn adapted to (P1 , . . , Pr ). On the other hand, by iterating (n−1) times the operation PROJ, we arrive to a ﬁnite family of polynomials in one variable X1 . d. of R adapted to this family: the real roots of the polynomials in the family cut the line in ﬁnitely many points and open intervals. 20 For every ﬁnite family P1 , . . , Pr in R[X1 , . . d. of Rn . d. of R3 adapted to the polynomial P = X 2 + Y 2 + Z 2 − 1. The Sylvester matrix of P and ∂P/∂Z is 1 0 X2 + Y 2 − 1 0 0 2 .

The quotient ring P(S) = R[X1 , . . , Xn ]/I(S) is called the ring of polynomial functions on S. Indeed, it can be identiﬁed with the ring of functions S → R which are the restriction of a polynomial. A nonempty 58 CHAPTER 3. TRIANGULATION OF SEMIALGEBRAIC SETS algebraic set V is said to be irreducible if it cannot be written as the union of two algebraic sets strictly contained in V . If V is irreducible, P(V ) is an integral domain, and we shall denote by K(V ) its ﬁeld of fractions (the ﬁeld of rational fractions on V ).