Cubic diophantine inequalities for split forms

TL;DR

This paper establishes bounds on the minimum number of variables needed for cubic forms that split into r parts to take arbitrarily small nonzero integral values, advancing understanding of Diophantine inequalities.

Contribution

It provides explicit bounds for s_0^{(r)} for r up to 6, extending previous results on cubic Diophantine inequalities for split forms.

Findings

Bounded s_0^{(r)} for r ≤ 6

Extended the range of r for which bounds are known

Improved understanding of small value representations of split cubic forms

Abstract

Denote by $s_0^{(r)}$ the least integer such that if $s \ge s_0^{(r)}$, and $F$ is a cubic form with real coefficients in $s$ variables that splits into $r$ parts, then $F$ takes arbitrarily small values at nonzero integral points. We bound $s_0^{(r)}$ for $r \le 6$.