A new method for computing asymptotics of diagonal coefficients of multivariate generating functions

Let $\sum_{\mathbf{n} \in \mathbb{N}^d} F_{\mathbf{n}} \mathbf{x}^{\mathbf{n}}$ be a multivariate generating function that converges in a neighborhood of the origin of $\mathbb{C}^d$. We present a new, multivariate method for computing the asymptotics of the diagonal coefficients $F_{a_1n,\ldots,a_dn}$ and show its superiority over the standard, univariate diagonal method. Several examples are given in detail