To be precise, $G({\ensuremath{\mathbf{z}}},t)$ and $H({\ensuremath{\mathbf{z}}},t)$ are defined as the numerator and denominator of Equation .

For the walks defined by ${\ensuremath{\mathcal{S}}}=\{e_1, -e_1, \dots, e_d, -e_d\}$, we have $$\frac{G({\ensuremath{\mathbf{z}}},t)}{H({\ensuremath{\mathbf{z}}},t)} = \frac{(1+z_1)\cdots(1+z_d)}{1 - t\sum_{k=1}^n (1+z_k^2)(z_1\cdots z_{k-1}z_{k+1}\cdots z_d)}.$$ Note that this rational function is not unique, in the sense that there are other rational functions whose diagonals yield the same counting sequence.

The singular variety associated to the kernel
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Here, we pause to note that the *combinatorial* symmetries of the step sets that we consider affect the *geometry* of the variety of $H({\ensuremath{\mathbf{z}}},t)$ – called the *singular variety*. This has a direct impact on both the asymptotics of the counting sequence under consideration and the ease with which its asymptotics are computed. In particular, any factors of the form $(1-z_k)$ present in the denominator of this rational function before simplification could have given rise to non-simple poles and thus made the singular variety non-smooth. Although non-smooth varieties can be handled in many cases – see [@PeWi13] – having a smooth singular variety is the easiest situation in which one can work in the multivariate setting. Understanding the interplay between the step set symmetry and the singular variety geometry, and in the 