Consider a convex rational polyhedral cone $C\subset\mathbb R^m$ with vertex at the origin. Let $X$ be the corresponding affine toric variety, i.e. $\mathbb C[X]=\mathbb C[\mathbb Z^m\cap C^\circ]$. Next consider the multivariate Hilbert series of $X$ $$S=\sum_{a\in\mathbb Z^m\cap C^\circ} x_1^{a_1}\ldots x_m^{a_m}.$$
Each summand in this series can be viewed as a function from $\mathbb C[X]$ which lets us speak of convergence in points of $X$. In fact, one may show that $S$ converges absolutely within a (metrically) open subset of $X$ to a rational function $\sigma\in\mathbb C(X)$. My question is: what is known about the geometric properties of $\sigma$ and, in particular, about its set of zeros?
