I'm trying to find out where a multivariate power series of the form
$\sum_{\alpha_1,...,\alpha_m \in \mathbb{N}}c_{\alpha_1} \cdots c_{\alpha_m}z_1^{\alpha_1} \cdots z_m^{\alpha_m}$
converges. I know that the Cauchy-Hadamard-Theorem can be applied here, however it would be more practical to work with a multivariate version of the ratio test. Does this exist, and if it does, where can I find out more about it?
Also, I have a (probably dumb) question about notation in the Cauchy-Hadamard-Theorem.
Wikipedia says,
Let $\alpha$ be a multi-index (a ''n''-tuple of integers) with $|\alpha|=\alpha_1+\cdots+\alpha_n$, then $f(x)$ converges with radius of convergence $\rho$ (which is also a multi-index) if and only if
$\lim_{|\alpha|\to\infty} \sqrt[|\alpha|]{|c_\alpha|\rho^\alpha}=1$
How exactly is $|c_\alpha|\rho^\alpha$ defined?
Is it $|c_{\alpha_1}+\cdots+c_{\alpha_m}| \cdot (\rho^{\alpha_1} \cdots \rho^{\alpha_m})$?