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

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

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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. Note that if x is not a root of P , the multiplicity of x as a root of P is equal to 0. 2. Let K be a field of characteristic zero. The element x ∈ K is a root of P ∈ K[X] of multiplicity µ if and only if P (µ)(x) 0, P (µ−1)(x) = = P (x) = P (x) = 0.

Let C = A B = c p+ q X p+ q + + c0. It is clear that all the coefficients of C are positive. It remains to prove that c2k ck−1 ck+1. 2 Real Root Counting Using the partition of (h, j) ∈ Z2 h > j and {(h, h − 1) h ∈ Z}. 49 in (j + 1, h − 1) ∈ Z2 h F F j F c2k − ck−1 ck+1 = ah a j bk−h bk− j + h j ah a j bk−h bk− j h>j − ah a j bk−h+1 bk− j −1 − h j = ah a j bk−h bk− j + h j a j +1 ah−1 bk− j −1 bk −h−1 h + ah a j bk −h+1 bk −j −1 h> j j ah ah−1 bk−h bk−h+1 − ah ah−1 bk−h+1bk−h h h − ah a j bk−h+1 bk− j −1 − h = j a j +1 ah−1bk −j bk−h h j (ah a j − ah−1 a j +1) (bk −j bk−h − bk− j −1 bk −h+1).

There are four cases to consider. 46 2 Real Closed Fields If ν is odd, 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), Var(Der(P ); d ) = Var(Der(P ); d ). 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 ).

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