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)
Note that PRem(P y , Q y) = PRem(P , Trui(Q)) y where degX (Qy) = i. Given a leaf L of TRems(P , Q), we denote by BL the unique path from the root of TRems(P , Q) to the leaf L. If N is a node in BL which is not a leaf, we denote by c(N ) the unique child of N in B L. 19. The Reali(C L) partition Ck. Moreover, y ∈ Reali(C L) implies that the signed remainder sequence of P y and Q y is proportional (up to a square) to the sequence of polynomials Pol(N ) y in the nodes along the path BL leading to L.
From 1 to −1) of P Q when x varies from a to b. It is clear that n−+ − n+− = σ(b) if σ(a) σ(b) = −1 0 if σ(a) σ(b) = 1. It follows from the deﬁnition of Cauchy index that Ind(Q/P ; a, b) + Ind(P /Q; a, b) = n−+ − n+−. Noting that Ind(R/Q; a, b) = Ind(P /Q; a, b), the claim of the lemma is now clear. 58: We can assume without loss of generality that a and b are not roots of a polynomial in the signed remainder sequence. Indeed if a < a < b < b with (a, a ] and [b , b) containing no root of the polynomials in the signed remainder sequence, it is clear that Ind(Q/P ; a, b) = Ind(Q/P ; a , b ).
23 easily implies the following theorem, known as the transfer principle for algebraically closed ﬁelds. 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.
Algorithms in Real Algebraic Geometry, Second Edition (Algorithms and Computation in Mathematics) by Saugata Basu, Richard Pollack, Marie-Francoise Roy,