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)
Proof: Obvious, using the decomposition of P in products of linear factors and polynomials of degree 2 with complex conjugate roots, since the product of two polynomials whose coeﬃcients are all non-negative have coeﬃcients that are all non-negative. The second case we consider is the case of normal polynomials. A polynomial A = a p X p + + a0 with non-negative coeﬃcients is normal if a) a p > 0, b) a2k ak−1 ak+1 for all index k, c) ah > 0 and a j > 0 for indices j < h implies a j+1 > 0, (with the convention that ai = 0 if i < 0 or i > p).
It is also called the Lefschetz Principle. 26. [Lefschetz principle] Suppose that C is an algebraically closed ﬁeld which contains the algebraically closed ﬁeld C. 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.
4. Let P = a p X p + + a0, a p 0, be a polynomial |a | with coeﬃcients in an ordered ﬁeld F. If |x| is bigger than 2 0≤i≤ p |a i | , p then P (x) and a p x p have the same sign. Proof: Suppose that |x| > 2 0≤i≤p which implies |x| > 2. Since P (x) = 1+ ap xp P (x) ≥ 1− ap xp 0≤i≤ p−1 ai i− p x , ap 0≤i≤ p−1 ≥ 1− 0≤i≤ p ai , ap |ai | |x|i− p |a p | |ai | |a p | (|x|−1 + |x|−2 + + |x|−p) 1 ≥ 1 − (1 + |x|−1 + + |x|− p+1) 2 1 1 − |x|−p > 0. = 1− 2 1 − |x|−1 We now examine a particular way to order the ﬁeld of rational functions R(X).
Algorithms in Real Algebraic Geometry, Second Edition (Algorithms and Computation in Mathematics) by Saugata Basu, Richard Pollack, Marie-Francoise Roy,