A note on the maximal rank

TL;DR

This paper establishes an upper bound on the X-rank of points relative to a variety, particularly when sub-generic X-rank points form a hypersurface, improving upon previous bounds with sharp examples.

Contribution

It introduces a new upper bound for X-rank in specific geometric configurations and demonstrates its sharpness through examples, advancing understanding of rank bounds.

Findings

New upper bound for X-rank established

Bound is sharp in certain cases

Improves upon existing bounds

Abstract

We give an upper-bound for the $X$-rank of points with respect to a non-degenerate irreducible variety $X$ in the case that sub-generic $X$-rank points generate a hypersurface. We give examples where this bound is sharp and it improves the existing ones.