Bivariate polynomial multiplication patterns

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 less than or equal to n, j less than or equal to m, e. g. with nz = 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 we mod (x(n+1), y(n+1)) admits the reduced upper bound 3n(2)+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(n+1)). There remains the issue to find the exact complexities of these multiplication problems.