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In this paper, we consider computations involving polynomials with inexact coefficients, i.e. with bounded coefficient errors. The presence of input errors changes the nature of questions traditionally asked in computer algebra. For instance, given two polynomials, instead of trying to compute their greatest common divisor, one might now try to compute a pair of polynomials with a non-trivial common divisor close to the input polynomials. We consider the problem of finding approximate common divisors in the context of inexactly specified polynomials. We develop efficient algorithms for the so-called nearest common divisor problem and several of its variants.

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In this paper, we consider the problem of interpolating univariate polynomials over a eld of characteristic zero that are sparse in (a) the Pochhammer basis or, (b) the Chebyshev basis. The polynomials are assumed to be given by black boxes, i.e., one can obtain the value of a polynomial at any point by querying its black box. We describe e cient new algorithms for these problems. Our algorithms may be regarded as generalizations of Ben-Or and Tiwari's (1988) algorithm (based on the BCH decoding algorithm) for interpolating polynomials that are sparse in the standard basis. The arithmetic complexity of the algorithms is O(t 2 + t log d) which is also the complexity of the univariate version of the Ben-Or and Tiwari algorithm. That algorithm and those presented here also share the requirement of 2t evaluation points.

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