Universality for Taylor coefficients of rational functions under perturbations

We introduce computational methods for analytic combinatorics in several variables. Our findings pertain to rational generating functions whose dominant singularity satisfies either of two conditions, one of which frequently holds for recursions arising from cluster algebras. We show that the coefficients are determined asymptotically by the leading homogeneous term of the denominator of the function near the dominant singularity. Applying this to various statistical physical models, we show their asymptotic behavior to be given by elliptic and hyperelliptic integrals, computation of which is already implemented in computer algebra packages.