By Saugata Basu, Richard Pollack, Marie-Francoise Roy,

ISBN-10: 3540330984

ISBN-13: 9783540330981

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**Extra resources for Algorithms in Real Algebraic Geometry, Second Edition (Algorithms and Computation in Mathematics)**

**Example text**

Let P ∈ K[X], of degree k, and x1, , xk be the roots of P (counted with multiplicities) in an algebraically closed ﬁeld C containing K. If a polynomial Q(X1, , Xk) ∈ K[X1, , Xk] is symmetric, then Q(x1, , xk) ∈ K. Proof: Let ei, for 1 ≤ i ≤ k, denote the i-th elementary symmetric function evaluated at x1, , xk. 12 gives ei ∈ K. 13, there exists R(T1, , Tk) ∈ K[T1, , Tk] such that Q(X1, Thus, Q(x1, , xk) = R(e1, , Xk) = R(E1, , Ek). , ek) ∈ K. 11. 11: a) ⇒ b) Let P ∈ R[X] a monic separable polynomial of degree p = 2m n with n odd.

9. Let C be a proper cone of F. If −a∈C, then C[a] = {x + a y x, y ∈ C } F is a proper cone of F. Proof: Suppose −1 = x + a y with x, y ∈ C. If y = 0 we have −1 ∈ C which is impossible. If y 0 then −a = (1/y 2) y (1 + x) ∈ C , which is also impossible. 8: Since the union of a chain of proper cones is a proper cone, Zorn’s lemma implies the existence of a maximal proper cone C which contains C. It is then suﬃcient to show that C ∪ −C = F, and to deﬁne x ≤ y by y − x ∈ C. Suppose that −a∈C. 9, C[a] is a proper cone and thus, by the maximality of C, C = C[a] and thus a ∈ C.

44. If A is normal and x > 0, then Var(A (X − x)) = 1. Proof: We can suppose without loss of generality that that 0 is not a root of A, that it that all the coeﬃcients of A are positive. Then a p−1 a p−2 a0 , ap a p−1 a1 and a p−1 a p−2 a0 −x −x − x. ap a p−1 a1 Since a p > 0 and -a0 x<0, the coeﬃcients of the polynomial (X − x) A = ap X p+1 + a p a p−1 − x Xp + ap + a1 a0 − x X − a0 x. a1 have exactly one sign variation. 35), is to interpret the even diﬀerence Var(Der(P ); a, b) − num(P ; (a, b]).

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