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Download Algorithms in Real Algebraic Geometry, Second Edition by Saugata Basu, Richard Pollack, Marie-Francoise Roy, PDF

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

Given a and b, we denote by v(P ; (a, b]) the number of virtual roots of P in (a, b] counted with virtual multiplicities. 47. v(P ; (a, b]) = Var(Der(P ); a, b). 47. 48. Let c be a root of P of virtual multiplicity v(P , c) ≥ 0. If no P (k) , 0 ≤ k < p has a root in [d, c), then v(P , c) = Var(Der(P ); d, c). Proof: The proof of the claim is by induction on p = deg(P ). The claim obviously holds if p = 0. Let w = v(P , c). − If c is a root of P , the virtual multiplicity of c as a root of P is w − 1.

The claim is obviously true for deg(P ) = 1. Suppose that the claim is true for every polynomial of degree < d. Since P = (X − x) µ−1 (µ Q + (X − x) Q ), and µ Q(x) 0, by induction hypothesis, P (x) = = P (µ−1)(x) = 0, P (µ)(x) 0. Conversely suppose that P (x) = P (x) = = P (µ−1)(x) = 0, P (µ)(x) 0. 1 (Taylor’s formula) at x, P = (X − x) µ Q, with Q(x) = P (µ)(x)/µ! 0. A polynomial P ∈ K[X] is separable if the greatest common divisor of P and P is an element of K \ {0}. A polynomial P is square-free if there is no non-constant polynomial A ∈ K[X] such that A2 divides P .

51. As a consequence, we can decide whether P has a root in R by checking whether Var(SRemS(P , P ); −∞, +∞) > 0. 50 (Sturm’s theorem). 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.

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