Rational approximation on cubic hypersurfaces

TL;DR

This paper calculates the approximation constant for rational points on smooth cubic hypersurfaces over number fields, assuming the presence of a rational line, and verifies the Coba conjecture for such hypersurfaces.

Contribution

It provides a general method to compute approximation constants on cubic hypersurfaces with rational lines and confirms the Coba conjecture in this context.

Findings

Computed the approximation constant for rational points on cubic hypersurfaces.

Verified the Coba conjecture for smooth cubic hypersurfaces with rational lines.

Established conditions under which the approximation constant can be explicitly determined.

Abstract

We compute the constant of approximation for an arbitrary rational point on an arbitrary smooth cubic hypersurface $X$ over a number field $k$, provided that there is a $k$-rational line somewhere on $X$. In the process, we verify the Coba conjecture for $X$.