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[80][81][82][83][84][85][86][87][88][89][90]. The final version of the paper includes a rather detailed analysis of the computing time of the algorithm, which was omitted from the preliminary version In addition, the final version describes several significant improvements to the algorithm of the earlier version.The paper describes a new quantifier elimination algorithm for the first order theory of real closed fields (equivalently, the first order theory of the ordered field of all real numbers--with addition, multiplication and order). The original quantifier elimination method for this theory, based on a generalization of Sturm's theorem, was published by A. Tarski in 1948 (although it was discovered by him about 1930}. Other methods were described by A.

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Given a set of r-variale integral polynomials, a cylindrical algebraic decomposition (cad) of euclidean r-space E T partitions ET into connected subsets compatible with the zeros of the polynomials. Collins (1975) gave an algorithm for cad construction as part of a new decision procedure for real closed fields. This algorithm has since been implemented and applied to diverse problems (optimization, curve display). New applications of it have been proposed (program verification, motion planrnng), Part I of the present paper has several purposes, FirsL, it provides an exposition of the essential aspects of the algorithm. Second, it corrects mi.p.or errors in the 1975 paper, and develops certain concepts introduced there. Third. it provides a framework fOI" the adjacency algorithm presented in Part n. ]n addition. it surveys the applications of cad's und provides a detailed example of the operation of the algorithm.Key\'iO,'ds: polynomial zeros, computer .....lgebru. computational geometry, semi-algebraic geometry. real closed fields, decision procedures, real algebraic geometry.

ABSTRACT. Let 9 be an integral domain, (9(9) the integral domain of polynomials over 9. L~ P, Q E (P(9) with m = deg (P) _> n = deg (Q) > 0. Let M be the matrix whose determina~ defines the resultant of P and Q. Let M~j be the submatrix of M obtained by deleting the la~ j rows of P coefficients, the last j rows of Q coeilicients and the last 2j+1 columns, exceptin cohlmn m + n -i -j (0 < i < j < n). The polynomial Rj(x) = ~.0 det (Mzj)x i is the j-t subresultant of P and Q, R0 being the resultant. If b = ~(Q), the leading coefficient of Q, ther exist uniquely R, S E (P(9) such that b~-'~+'P = QS + R with deg (R) < n; define ~(P, Q) = Define Pi E (P (5: (1) P~ 6 (P(9) for 1 < i < k; (2) P~ ~ ±AkR,~_~-~, whet A~ = .t~,-~,' Il~:-L''-~(~-~) ", (3) c~-~-lP~ = ~A~+~R,~.~ ," (4) R~ = 0 for n~ < j < n~_~ -1. Takinl 9 to be the integers [, or (P~(I), these results provide new algorithms for computing resultant or greatest common divisors of univariate or multivariate polynomials. Theoretical analysi~ and extensive testing on a high-speed computer show the new g.c.d, algorithm to be faste than known algorithms by a large factor. When applied to bivariate polynomials, for example this factor grows rapidly with the degree and exceeds 100 in practical cases.

Uspensky's 1948 book on the theory of equations presents an algorithm, based on Descartes' rule of signs, for isolating the real roots of a squarefree polynomial with real coefficients. Programmed in SAC-I and applied to several classes of polynomials with integer coefficients, Uspensky's method proves to be a strong competitor of the recently discovered algorithm of Collins and Loos. It is shown, however, that it's maximum computing time is exponential in the coefficient length. This motivates a modification of the Uspensky algorithm which is quadratic in the coefficient length and which also performs well in the practical test cases.