Asymptotics of Generating Coefficients along a Ray

Question

Suppose I have a multidimensional array of numbers $a(n_1,\ldots,n_r)$, for $n_1,\ldots,n_r\in\mathbb N\cup\{0\}$. I can form the generating function $$A(x_1,\ldots,x_r)=\sum_{n_1,\ldots,n_r\geq 0}a(n_1,\ldots,n_r)x_1^{n_1}\cdots x_r^{n_r}.$$ I am interested in computing the asymptotics of $a(n_1,\ldots,n_r)$ along a ray. More precisely, I fix constants $c_1,\ldots,c_r\in[0,1]$ and consider the asymptotics of the sequence $b(n)=a(\left\lfloor c_1n\right\rfloor,\ldots,\left\lfloor c_rn\right\rfloor)$. I actually only care about the exponential growth rate $\lim_\limits{n\to\infty}b(n)^{1/n}$. Is it possible to obtain this information from the generating function $A(x_1,\ldots,x_r)$? I know there have been recent developments in multivariate analytic combinatorics when the generating function $A(x_1,\ldots,x_r)$ is rational, but I am wondering if I can still say something if the generating function is, say, algebraic. The specific generating function I have in mind has a square root in it.