On computing greatest common divisors with polynomials given by black boxes for their evaluations

The black box representation of a multivariate polynomial is a function that takes as input a value for each variable and then produces the value of the polynomial. We revisit the problem of computing the greatest common divisor (GCD) in black box format of several multivariate polynomials that themselves are given by black boxes. To this end an improved version of the algorithm sketched in Kaltofen and Trager [J. Symbolic Comput., vol. 9, nr. 3, p. 311 (1990)] is described. Also the full analysis of the improved algorithm is given. Our algorithm constructs in random polynomial-time a procedure that will evaluate a fixed associate of the GCD at an arbitrary point (supplied as its input) in polynomial time. The randomization of the black box construction is of the Monte-Carlo kind, that is with controllably high probability the procedures evaluating the GCD are correct at all input points. Finally, a Maple prototype implementation as well as our plans for developing a subsystem for manipulating multivariate polynomials and rational functions in black box representation are presented.

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