Real Rank with Respect to Varieties

TL;DR

This paper investigates the maximal real rank of points relative to real varieties, establishing bounds, constructing tight examples, and exploring differences between real and complex ranks, with applications to tensor decompositions and algebraic geometry.

Contribution

It provides bounds on maximal real rank based on codimension, constructs examples where these bounds are tight, and proves a conjecture on symmetric tensor ranks.

Findings

Maximal real rank can be bounded by the codimension of the variety.

Existence of varieties where the bound on real rank is tight.

The gap between real and complex ranks can be arbitrarily large.

Abstract

We study the real rank of points with respect to a real variety $X$. This is a generalization of various tensor ranks, where $X$ is in a specific family of real varieties like Veronese or Segre varieties. The maximal real rank can be bounded in terms of the codimension of $X$ only. We show constructively that there exist varieties $X$ for which this bound is tight. The same varieties provide examples where a previous bound of Blekherman-Teitler on the maximal $X$-rank is tight. We also give examples of varieties $X$ for which the gap between maximal complex and the maximal real rank is arbitrarily large. To facilitate our constructions we prove a conjecture of Reznick on the maximal real symmetric rank of symmetric bivariate tensors. Finally we study the geometry of the set of points of maximal real rank in the case of real plane curves.