Lazy and Forgetful Polynomial Arithmetic and Applications

Abstract

Abstract. We present lazy and forgetful algorithms for multiplying and dividing multivariate polynomials. The lazy property allows us to compute the i-th term of a polynomial without doing the work required to compute all the terms. The forgetful property allows us to forget earlier terms that have been computed to save space. For example, given polynomials A, B, C, D, E we can compute the exact quotient Q = A×B−C×D E without explicitly computing the numerator A × B − C × D which can be much larger than any of A, B, C, D, E and Q. As applications we apply our lazy and forgetful algorithms to reduce the maximum space needed by the Bareiss fraction-free algorithm for computing the determinant of a matrix of polynomials and the extended Subresultant algorithm for computing the inverse of an element in a polynomial quotient ring.

Multivariate Power Series in Maple

“…The usefulness of lazy evaluation in computer algebra has been studied for a few decades. In particular, see the work of Karczmarczuk [10], discussing different mathematical objects with an infinite length; Burge and Watt [7], and van der Hoeven [17], discussing lazy univariate power series; and Monagan and Vrbik [12], discussing lazy arithmetic for polynomials.…”