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

In this paper, we examine the problem of computing the greatest common divisor (GCD) of univariate polynomials represented in different bases. When the polynomials are represented in Newton basis or a basis of orthogonal polynomials, we show that the well-known Sylvester matrix can be generalized. We give fraction-free and modular algorithms to directly compute the GCD in the alternate basis. These algorithms are suitable for computation in domains where growth of coefficients in intermediate computations are a central concern. In the cases of Newton basis and bases using certain orthogonal polynomials, we also show that the standard subresultant algorithm can be applied easily. If the degrees of the input polynomials is at most n and the degree of the GCD is at least n/2, our algorithms outperform the corresponding algorithms using the standard power basis.

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“…It would be interesting to see if our results can be applied to "black box polynomials" as well [15,22].…”

Section: Discussionmentioning

confidence: 99%

On computing polynomial GCDs in alternate bases

Cheng

Labahn

2006

Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation

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“…A parallel version of the Monte Carlo-randomized algorithm in [18] is used to construct a black box for evaluating the GCD of f l , . .…”

Section: A Monte Carlo Gcd Algorithmmentioning

confidence: 99%

“…, wT2 E Gf(pM -Probability and Parallel Running Time Analysis fails. Then, from [18], we know that 4 4 3d2/pM. algorithm is incorrect.…”

Section: Algorithm Conuerting the Black Box Into The Sparse Representmentioning

confidence: 99%

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Interpolation of Sparse Multivariate Polynomials over Large Finite Fields with Applications

Huang

Rao

1999

Journal of Algorithms

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“…This is based, of course, on resultant theory ( [2], [3]), and also works for multivariate polynomials. So for the polynomial case, we expect that with only about one calculation of a pairwise gcd of two polynomials we find the true gcd of many polynomials, provided that A is chosen large enough.…”

Section: Fact 2 Letmentioning

confidence: 99%