Let $R$ be a commutative ring. Let $f(x_1, \dots, x_n), g(x_1, \dots, x_n)$ be two multivariate polynomials with the same $x_i$-terms with maximal total degree $\delta$, but with different coefficients in $R$. How fast can we compute the product of $f$ and $g$, i.e. the resulting coefficients of each term?
For univariate multiplication in a general setting like this, one can use variants of schonhage-strassen to achieve $O(\delta \log \delta \log\log \delta)$ (e.g. Theorem 8.23 in von zur Gathen & Gerhard, Modern Computer Algebra).
I have only been able to find a non-trivial algorithm for when $R$ is a field of characteristic 0, but not for $R$ being a general commutative ring.