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We consider asymptotics of power series coefficients of rational functions ofthe form $1/Q$ where $Q$ is a symmetric multilinear polynomial. We review anumber of such cases from the literature, chiefly concerned either withpositivity of coefficients or diagonal asymptotics. We then analyze coefficientasymptotics using ACSV (Analytic Combinatorics in Several Variables) methods.While ACSV sometimes requires considerable overhead and geometric computation,in the case of symmetric multilinear rational functions there are somereductions that streamline the analysis. Our results include diagonalasymptotics across entire classes of functions, for example the general3-variable case and the Gillis-Reznick-Zeilberger (GRZ) case, where thedenominator in terms of elementary symmetric functions is $1 - e_1 + c e_d$ inany number $d$ of variables. The ACSV analysis also explains a discontinuousdrop in exponential growth rate for the GRZ class at the parameter value $c =(d-1)^{d-1}$, previously observed for $d=4$ only by separately computingdiagonal recurrences for critical and noncritical values of $c$.

Diagonal asymptotics for symmetric rational functions via ACSV