By K. Ueno

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Hence any bounded connected component of A contains at least a component of {p > 0} and ﬁnally: B0 (A) B0 ({p > 0}). 4 Semialgebraic and Tame Sets 51 The components of {p > 0} are open, hence the image by p of such a component is a non trivial interval of the type ]0, C[ or ]0, C] (C ∈ R+ ∪{∞}). By Sard’s theorem, let η > 0 be a suﬃciently small regular value of p such that in each bounded component of {p > 0} there is at least one component Zα of the regular hypersurface {p = η} = Z. Now take a generic linear form l on Rn .

Are related to the degree, whereas the localization of these data, or local invariants, are related to the multiplicity. 2 and of the deﬁnition of the density, that a semialgebraic subset A ⊂ Rn has its density bounded by its complexity, namely: Θ (A0 ) c(n, k)B0,n− . 5. Let A0 be a germ of a tame set of dimension at the origin of Rn . The following upper bound for the density Θ (A0 ) of A0 holds: Θ (A0 ) Proof. We just notice that Θ (σP ) B0,n− . B0,n− . Remark. We can ﬁnd a better bound for Θ (A0 ), say Θ (A0 ) β0,n− , where β0,n− has the same nature as B0,n− , the multiplicity mij of the polynomials pij replacing the degrees dij .

Dn , respectively. ,n equals n) is bounded by di . ,n Proof. Consider the complexiﬁcation of equations (∗) on Cn and notice that a nondegenerate real zero x of (∗) is also a nondegenerate solution of the complexiﬁed system. Hence the inequality follows from the complex Bezout theorem (see [Ben-Ris] for instance). Remark. The above inequality is not true in general for isolated, but possibly degenerate, solutions of (∗). For instance, let us consider the following system: f1 (x1 , . . , xn−1 ) = 0 ..