Small solutions to homogeneous and inhomogeneous cubic equations

We study the solubility of cubic equations over the integers. Assuming a necessary congruence condition, the existence of such solutions is established when the $h$-invariant of $C$ is at least $14$, improving on work of Davenport-Lewis and generalizing the method from Heath-Brown's seminal work in the homogeneous case. We also provide an upper bound on the smallest solution, polynomially in the height of the coefficients. The method also yields new results in the homogeneous case where we generalize and improve on previous work of Browning, Dietmann and Elliott.