Typical and Admissible ranks over fields

TL;DR

This paper introduces the concept of admissible rank for points relative to a real algebraic variety, exploring its properties, existence, and differences from traditional complex rank, especially in the context of rational normal curves.

Contribution

It defines admissible rank and labels, analyzes their properties for generic points, and provides examples where admissible rank differs from complex rank.

Findings

Admissible rank can be strictly larger than complex rank.

For rational normal curves, a label always exists for generic points.

Examples show cases where admissible rank does not exist or exceeds complex rank.

Abstract

Let $X(\RR)$ be a geometrically connected variety defined over $\RR$ and such that the set of all its (also complex) points $X(\CC)$ is non-degenerate. We introduce the notion of \emph{admissible rank} of a point $P$ with respect to $X$ to be the minimal cardinality of a set of points of $X(\CC)$ such that $P\in \langle S \rangle$ that is stable under conjugation. Any set evincing the admissible rank can be equipped with a \emph{label} keeping track of the number of its complex and real points. We show that in the case of generic identifiability there is an open dense euclidean subset of points with certain admissible rank for any possible label. Moreover we show that if $X$ is a rational normal curve than there always exists a label for the generic element. We present two examples in which either the label doesn't exists or the admissible rank is strictly bigger than the usual complex rank.