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Efficient algorithms are derived for computing the entries of the Bezout resultant matrix for two univariate polynomials of degree n and for calculating the entries of the Dixon-Cayley resultant matrix for three bivariate polynomials of bidegree (m, n). Standard methods based on explicit formulas require O(n 3 ) additions and multiplications to compute all the entries of the Bezout resultant matrix. Here we present a new recursive algorithm for computing these entries that uses only O(n 2 ) additions and multiplications. The improvement is even more dramatic in the bivariate setting. Established techniques based on explicit formulas require O(m 4 n 4 ) additions and multiplications to calculate all the entries of the Dixon-Cayley resultant matrix. In contrast, our recursive algorithm for computing these entries uses only O(m 2 n 3 ) additions and multiplications.

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Macaulay's concise but explicit expression for nmltivariate resultants has many potential applications in computer-aided geometric design. Here we describe its use in solid modeling for finding the intersections of three implicit quadric surfaces. By B6zout's theorem, three quadric surfaces have either at most eight or intlnitely many intersections. Our method finds the intersections, when there are finitely many, by generating a polynomial of degree at most eight whose roots are the intersection coordinates along an appropriate axis. Only addition, subtraction, and multiplication are required to find the polynomial. But when there are pmsibilities of extraneous roots, division and greatest common divisor computations are necessary to identify and remove them.

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