The divisor function along sums of two biquadrates

We establish power saving asymptotics for the sum of the divisor function along a binary quartic form, improving on work of Daniel. The proof involves an application of a recent two dimensional delta method due to Li, Rydin-Myerson, and Vishe and an exploitation of $\mathrm{GL}_2$ automorphic forms arising from the factorization of varying cubic Dedekind zeta functions.