Small solutions to homogeneous and inhomogeneous cubic equations

TL;DR

This paper advances the understanding of integer solutions to cubic equations by establishing solubility under certain conditions, providing bounds on the smallest solutions, and improving previous results in both homogeneous and inhomogeneous cases.

Contribution

It introduces new methods to prove solubility of cubic equations with high $h$-invariant and offers improved bounds on minimal solutions, extending prior work in the field.

Findings

Established solubility for cubic equations with $h$-invariant ≥ 14.

Provided polynomial bounds on the smallest solutions relative to coefficients.

Generalized and improved previous results for homogeneous cubic equations.

Abstract

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.