By A. Campillo

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L with x>.. xl-' E R[i]. L, hand h', h"l- h',such that X>.. xl-' E R[i], X>.. xl-' E R[i]. This implies that x>.. + xI-' E R[i] and x>"xl-' E R[i]. These elements x>.. and xI-' 32 2 Real Closed Fields are the solutions of a quadratic equation with coefficients in R[i], which has its two solutions in R[i], since the classical method for solving polynomials of degree 2 works in R[i] when R is real closed. The polynomial P thus has a root in R[i]. For P = apXP + ... + ao E R[i][X], we write P = apXP + ...

44. Finally, V(P) = O. 2 The Cauchy Index Let P be a non-zero polynomial with coefficients in areal closed field R. Not only would we like to determine whether P has a root in R but also to determine whether P has a root at which another polynomial Q is positive. With this goal in mind, it is profitable to look at the jumps (discontinuities) of the rational function P(c) = 0, Q(c) I- o. P~Q = iQ~cl + Re, If P~Q . L then where Re is a rational function defined at c. It is now 44 2 Real Closed Fields obvious that if Q(c) 0, then P'Q p jumps from -00 to +00 at c, and if pP'Q jumps from +00 to -00 at c.

Fig. 2. Graph of the rational function lnd (~ ja, b) ~. is equal to p if and only if q = p - 1, the signs of the leading coefficients of P and Q are equal, all the roots of P and Q are simple and belong to (a, b), and there is exactly a root of Q between two roots of P. b) If R = Rem(Q, P), it follows clearly from the definition that lnd (~ja,b) = lnd (;ja,b). With this definition we can reformulate our observation, using the following notation. 50. The Sturm-query of Q for P in (a, b) is the number SQ(Q,Pja,b) = #({x E (a,b) I P(x) = O/\Q(x) > O})#({XE (a,b) I P(x) =O/\Q(x)