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

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

Example text

2) If ν is odd, and sign(P (ν +1)(c) P (c)) < 0, Var(Der(P ); d) = Var(Der(P ); d), Var(Der(P ); c) = Var(Der(P ); c) + 1, Var(Der(P ); d ) = Var(Der(P ); d ) + 1. 3) If ν is even, and sign(P (ν +1)(c) P (c)) > 0, Var(Der(P ); d) = Var(Der(P ); d), Var(Der(P ); c) = Var(Der(P ); c), Var(Der(P ); d ) = Var(Der(P ); d ). 4) If ν is even, and sign(P (ν +1)(c) P (c)) < 0, Var(Der(P ); d) = Var(Der(P ); d) + 1, Var(Der(P ); c) = Var(Der(P ); c) + 1, Var(Der(P ); d ) = Var(Der(P ); d ) + 1. 5) The claim is true in each of these four cases.

The claim is clearly true if p = 0. Suppose that Taylor’s formula holds for p − 1: p−1 X p−1 = i=0 (p − 1)! x p−1−i (X − x)i. (p − 1 − i)! i! Then, since X = x + (X − x), p−1 X p = (x + (X − x)) p = i=0 since i=0 (p − 1)! x p−1−i (X − x)i (p − 1 − i)! i! p! x p−i (X − x)i (p − i)! i! p! (p − 1) p! = + . (p − i)! i! (p − i)! (i − 1)! (p − 1 − i)! (i − 1)! Hence, Taylor’s formula is valid for any polynomial using the linearity of derivation. Let x ∈ K and P ∈ K[X]. The multiplicity of x as a root of P is the natural number µ such that there exists Q ∈ K[X] with P = (X − x) µ Q(X) and Q(x) 0.

36. Let c be a root of P of multiplicity µ ≥ 0. If no P (k) , 0 ≤ k ≤ p, has a root in [d, c) ∪ (c, d ], then a) Var(Der(P ); d, c) − µ is non-negative and even, b) Var(Der(P ); c, d ) = 0. Proof: We prove the claim by induction on the degree of P . The claim is true if the degree of P is 1. Suppose ﬁrst that P (c) = 0, and hence µ > 0. By induction hypothesis applied to P , a) Var(Der(P ); d, c) − (µ − 1) is non-negative and even, b) Var(Der(P ); c, d ) = 0. The sign of P at the left of c is the opposite of the sign of P at the left of c and the sign of P at the right of c is the sign of P at the right of c.