Algorithms in Real Algebraic Geometry, Second Edition by Saugata Basu, Richard Pollack, Marie-Francoise Roy,

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)

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Real closed fields are characterized as follows. 11. If R is a field then the following properties are equivalent: a) b) c) d) R is real closed. R[i] = R[T ]/(T 2 + 1) is an algebraically closed field. R has the intermediate value property. R is a real field that has no non-trivial real algebraic extension, that is there is no real field R1 that is algebraic over R and different from R. 11. Let K be a field. A polynomial Q(X1, , Xk) ∈ K[X1, , Xk] is symmetric if for every permutation σ of {1, , k}, Q(Xσ(1), , Xσ(k)) = Q(X1, , Xk).

Using the notion of Cauchy index we can reformulate our preceding discussion, using the following notation. 56. [Tarski-query] Let P =0 and Q be elements of K[X]. The Tarski-query of Q for P in (a, b) is the number TaQ(Q, P ; a, b) = sign(Q(x)). x∈(a,b),P (x)=0 Note that TaQ(Q, P ; a, b) is equal to #({x ∈ (a, b) P (x) = 0 ∧ Q(x) > 0}) − #({x ∈ (a, b) P (x) = 0 ∧ Q(x) < 0}) F F where #(S) is the number of elements in the finite set S. The Tarski-query of Q for P on R is simply called the Tarski-query of Q for P , and is denoted by TaQ(Q, P ), rather than by TaQ(Q, P ; −∞, +∞).

Let P = a p X p + + a0 be a univariate polynomial in R[X]. We write Var(P ) for the number of sign variations in a0, , a p and pos(P ) for the number of positive real roots of P , counted with multiplicity. 33. [Descartes’ law of signs] − Var(P ) ≥ pos(P ) − Var(P ) − pos(P ) is even. 33 (Descartes’s law of signs) due to Budan and Fourier. 34. [Sign variations in a sequence of polynomials at a] Let P = P0, P1, , Pd be a sequence of polynomials and let a be an element of R ∪ {−∞, +∞}. 4). For example, if P = X 5, X 2 − 1, 0, X 2 − 1, X + 2, 1, Var(P; 1) = 0.

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