On the real \(X\)-ranks of points of \(\mathbb{P}^n(\mathbb{R})\) with respect to a real variety \(X\subset\mathbb{P}^n\)

Abstract

Let \(X\subset\mathbb{P}^n\) be an integral and non-degenerate \(m\)-dimensional variety defined over \(\mathbb{R}\). For any \(P \in \mathbb{P}^n(\mathbb{R})\) the real \(X\)-rank \(r_{X,\mathbb{R}}(P)\) is the minimal cardinality of \(S\subset X(\mathbb{R})\) such that \(P\in \langle S\rangle\). Here we extend to the real case an upper bound for the \(X\)-rank due to Landsberg and Teitler.