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

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52. Consider the polynomial P = X 4 − 5X 2 + 4. The Sturm sequence of P is SRemS0(P , P ) = P = X 4 − 5X 2 + 4, SRemS1(P , P ) = P = 4X 3 − 10X , 5 SRemS2(P , P ) = X 2 − 4, 2 18 X, SRemS3(P , P ) = 5 SRemS4(P , P ) = 4. The signs of the leading coefficients of the Sturm sequence are + + + + + and the degrees of the polynomials in the Sturm sequence are 4, 3, 2, 1, 0. The signs of the polynomials in the Sturm sequence at −∞ are + − + − + , and the signs of the polynomials in the Sturm sequence at +∞ are + + + + + , so Var(SRemS(P , P ); −∞, +∞) = 4.

40 relies on the following lemmas. 41. The polynomial X − x is normal if only if x 0. Proof: Follows immediately from the definition of a normal polynomial. 42. A quadratic monic polynomial A with complex conjugate roots is normal if and only if its roots belong to the cone B. Proof: Let a + i b and a − i b be the roots of A. Then A = X 2 − 2 a X + (a2 + b2) is normal if and only if a) − 2 a 0, b) a2 + b2 0, c) ( − 2 a)2 a2 + b2. that is if and only if a 0 and 4 a2 a2 + b2, or equivalently a + i b ∈ B.

11 and that the conjugate of a root of P is a root of P . 9. Prove that, in a real closed field, a second degree polynomial P = a X 2 + b X + c, a 0 has a constant non-zero sign if and only if its discriminant b2 − 4 a c is negative. Hint: the classical computation over the reals is still valid in a real closed field. 20. Let R be a real closed field, P ∈ R[X] such that P does not vanish in (a, b), then P has constant sign in the interval (a, b). 11. This proposition shows that it makes sense to talk about the sign of a polynomial to the right (resp.

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