In the single variable case, the power series $$\sum_{n=0}^\infty a_n z^n $$ defines an entire function, provided that $$R^{-1}:=\limsup_{n \to \infty} |a_n|^{1/n}=0. $$ Moreover, if $R^{-1} >0$, the series converges for $|z|<R$, and has some singularity on $|z|=R$.
I'd like to know what happens in the case of $n \geq 2$ variables: Using multi-index notation, a power series has the form $$\sum_{|\alpha| \geq 0} a_\alpha z^\alpha .$$ Is there a way to know, based on the coefficients $\{a_\alpha\}$, if the series defines an entire function? Is there any way to gain insights on the region of convergence at all?
Thank you!