By Coste M.

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C: AC,j = {(x , xk+1 ) ∈ C × R ; xk+1 = ξC,j (x )} , – or a band of the cylinder bounded from below and from above by the graphs of functions ξC,j and ξC,j+1 , for j = 0, . . , C , where we take ξC,0 = −∞ and ξi, C +1 = +∞: BC,j = {(x , xk+1 ) ∈ C × R ; ξC,j (x ) < xk+1 < ξC,j+1 (x )} . 3. d. is semialgebraically homeomorphic to an open hypercube (0, 1)d (by convention, (0, 1)0 is a point). Proof. We prove the property of the proposition for cells of Ck , by induction on k. The key point is to observe that, using the notation above, every graph AC,j is semialgebraically homeomorphic to C and every band BC,j is semialgebraically homeomorphic to C × (0, 1).

Let mi (resp. pi ) be the multiplicity of zi as a root of P (a , Xn ) (resp. Q(a , Xn )), where multiplicity zero means “not a root”. The degree of gcd(P (a , Xn ), Q(a , Xn )) is ki=1 min(mi , pi ), and each zi has multiplicity min(mi , pi ) as a root of this gcd. Choose ε > 0 such that all disks D(zi , ε) are disjoint. For every x ∈ C suﬃciently close to a , each disk D(zi , ε) contains a root of multiplicity mi of P (x , Xn ) and a root of multiplicity pi of Q(x , Xn ). Since the degree of gcd(P (x , Xn ), Q(x , Xn )) is equal to ki=1 min(mi , pi ), this gcd must have one root of multiplicity min(mi , pi ) in each disk D(zi , ε) such that min(mi , pi ) > 0.

D.. The proof is by induction on n. Z If n = 1, either C is a point and C is equal to this point, or C is a nonempty Z open interval and C = R. We have algebraic dimension 0 in the ﬁrst case and 1 in the second case. Let n > 1, and assume the theorem proved for n − 1. Let π : Rn → Rn−1 be the projection on the space of the ﬁrst n − 1 coordinates. , semialgebraically homeomorphic to Z (0, 1)d . By the inductive assumption, dim D = d (dimension as algebraic set). We have to consider two cases. 1.

### An introduction to semialgebraic geometry by Coste M.

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