By Saugata Basu, Richard Pollack, Marie-Francoise Roy,
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The aim of this ebook is to illustrate that advanced numbers and geometry should be combined jointly superbly. This ends up in effortless proofs and common generalizations of many theorems in aircraft geometry, akin to the Napoleon theorem, the Ptolemy-Euler theorem, the Simson theorem, and the Morley theorem.
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Extra info for Algorithms in Real Algebraic Geometry, Second Edition (Algorithms and Computation in Mathematics)
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.