Suppose you have a function of two variables, say $f(x,y)$, that is nice enough to equal a power series $\sum_{m,n} a_{m,n}x^my^n$ in some region about the origin.
Is there a rough asymptotic formula for $a_{m,n}$ based solely on $m$, $n$ and the singularities of $f$?
This question is essentially a two-dimensional version of the root test: if $f(x)=\sum_n a_nx^n$, and $r$ is the modulus of the smallest singularity of $f$, then $a_n\sim r^{-n}$.
Two simple-but-contrasting examples are $f_1(x,y)=\frac{1}{(1-2x)(1-3y)}$ with $a_{m,n}=2^m3^n$ and $f_2(x,y)=\frac{1}{1-xy}$ with $a_{m,n}=0$ unless $m=n$ and then $a_{m,m}=1$.
As a particular application, I am interested in approximating the coefficients of the rational function $\frac{1}{(1-x)^2(1-y)^2-xy}$, which is the generating function of an array of numbers of combinatorial objects I am studying.