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Geometry And Topology

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

Let P ∈ K[X], of degree k, and x1, , xk be the roots of P (counted with multiplicities) in an algebraically closed field C containing K. If a polynomial Q(X1, , Xk) ∈ K[X1, , Xk] is symmetric, then Q(x1, , xk) ∈ K. Proof: Let ei, for 1 ≤ i ≤ k, denote the i-th elementary symmetric function evaluated at x1, , xk. 12 gives ei ∈ K. 13, there exists R(T1, , Tk) ∈ K[T1, , Tk] such that Q(X1, Thus, Q(x1, , xk) = R(e1, , Xk) = R(E1, , Ek). , ek) ∈ K. 11. 11: a) ⇒ b) Let P ∈ R[X] a monic separable polynomial of degree p = 2m n with n odd.

61.

If Φ is a sentence in the language of fields with coefficients in C, then it is true in C if and only if it is true in C . 23, there is a quantifier free formula Ψ which is C-equivalent to Φ. 22 that Ψ is C -equivalent to Φ as well. Notice, too, that since Ψ is a sentence, Ψ is a boolean combination of atoms of the form c = 0 or c 0, where c ∈ C. Clearly, Ψ is true in C if and only if it is true in C . The characteristic of a field K is a prime number p if K contains Z/p Z and 0 if K contains Q. The meaning of Lefschetz principle is essentially that a sentence is true in an algebraic closed field if and only if it is true in any other algebraic closed field of the same characteristic.

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Algorithms in Real Algebraic Geometry, Second Edition (Algorithms and Computation in Mathematics) by Saugata Basu, Richard Pollack, Marie-Francoise Roy,


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