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This paper introduces singular value decomposition (SVD) algorithms for some standard polynomial comput at ions, in the case where the coefficients are inexact or imperfectly known. We first give an algorithm for computing univariate GCD'S which gives exact results for interesting nearby problems, and give efficient algorithms for computing precisely how nearby.We generalize this to multivariate GCD comput ation. Next, we adapt Lazard's u-resultant algorithm for the solution of overdetermined systems of polynomial equations to the inexact-coefficient case. We also briefly discuss an application of the modified Lazard's method to the location of singular points on approximately known projections of algebraic curves.

Algorithms are developed that adopt a novel implicit representation for multivariate polynomials and rational functions with rational coefficients, that of black boxes for their evaluation. We show that within this evaluation box representation the polynomial greatest common divisor and factorization problems as well as the problem of extracting the numerator and denominator of a rational function can be solved in random polynomial-time in the usual parameters. Since the resulting evaluation programs for the goal polynomials can be converted efficiently to sparse format, solutions to sparse problems such as the sparse rational interpolation problem follow as a consequence.

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