Bivariate polynomial multiplication patterns

Abstract

Motivated by multiplication of numerical univariate polynomials with various kinds of truncation we study corresponding bivariate problems A(x, y)·B(x, y) = C(x, y) in the algebraic setting with indeterminate coefficients over suitable ground fields, counting essential multiplications only. The rectangular case concerning factors A, B with entries x ⁱ y ^j for i ≤ n, j≤ m, e. g. with m = n, has complexity (2n + 1)². Here multiplication with single truncation, computing the product C(x,y) mod x ⁿ⁺¹, or mod y ⁿ⁺¹, seems still to have the same full multiplication complexity, as is well-known for the univariate case, while the double truncation case mod (x ⁿ⁺¹, y ⁿ⁺¹) admits the reduced upper bound 3n² + 4n + 1, opposed to a lower bound of 2n ² + 4n + 1. We have a similar saving factor for the triangular case with factors A, B of total degree n to be multiplied mod (x ⁿ⁺¹,x ⁿ y,...,y ^xn+1). There remains the issue to find the exact complexities of these multiplication problems.