process dealing with the non-smooth cases, is promising future work.

Analytic combinatorics in several variables {#sec:acsv}
===========================================

Following the work of Pemantle and Wilson [@PeWi02] and Raichev and Wilson [@RaWi08], we can determine the dominant asymptotics for the diagonal of the multivariate power series $
\frac{G({\ensuremath{\mathbf{z}}},t)}{H({\ensuremath{\mathbf{z}}},t)}$ by studying the variety (complex set of zeroes) ${\ensuremath{\mathcal{V}}}\subseteq \mathbb{C}^{d+1}$ of the denominator $$H({\ensuremath{\mathbf{z}}},t) = 1 - t\cdot(z_1\cdots z_d)S({\ensuremath{\mathbf{z}}}).$$ To begin, a particular set of singular points – called the *critical points* – containing all singular points which *could* affect the asymptotics of $\Delta (G/H)$ are computed in Section \[sec:crtpts\]. The set of critical points is then refined to those which determine the dominant asymptotics up to an exponential decay in Section \[sec:minpts\]; this refined set is called the set of *minimal points* as they are the critical points which are ‘closest’ to the origin in a sense made precise below. The enumerative results come from calculating a Cauchy residue type integral, and after determining the minimal points we determine asymptotics in Section \[sec:results\] using pre-computed formulas for such integrals which can be found in [@PeWi13]. In fact, up to polynomial decay there is only one singular point which determines dominant asymptotics for each 