We present alternative algorithms for computing symmetric powers of linear ordinary differential operators.Our algorithms are applicable to operators with coefficients in arbitrary integral domains and become faster than the traditional methods for symmetric powers of sufficiently large order, or over sufficiently complicated coefficient domains. The basic ideaa are also applicable to other computations involving cyclic vector techniques, such as exterior powers of differential or difference operators.
Abstract.Computing only the low degree terms of the product of two univariate polynomials is called a short multiplication. By decomposition into subproblems, a short multiplication can be reduced to appropriate addition of the results of a number of full multiplications. In this paper a new way of choosing the size of the subproblems is proposed. Computing the quotient of two polynomials is called a short division. The ideas used in the short multiplication algorithm are transferred to an algorithm for short divisions. Finally, several applications of short multiplications and divisions are pointed out.
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