I have a rational bivariate generating function $A(x,y)=\sum_{n\geq 1}\sum_{k\geq 1}a(n,k)x^ny^k$. I would like to find the exponential growth rate of $a(n,k)$ as $n,k\to\infty$ with $k\sim \delta n$ for some fixed $\delta\in(0,1)$. I'm wondering if there is a simple way to do this. I know that if I am working with univariate generating functions, I can obtain the exponential growth rate by looking at the absolute value of the pole of the generating function that is closest to the origin. It seems like something similar should be true for bivariate generating functions.

I should stress that I am not looking for precise asymptotics. Rather, I would like a simple way to obtain the growth rate $$\lim_{n\to\infty}\sqrt[n]{a(n,k)}$$ when $k\sim \delta n$ (say $k=\lfloor\delta n\rfloor$).


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