Real and complex rank for real symmetric tensors with low ranks

TL;DR

This paper investigates the relationship between real and complex tensor ranks for symmetric tensors, showing how differences in their minimal decompositions are geometrically constrained under certain rank sum conditions.

Contribution

It provides a geometric characterization of the difference between real and complex decompositions for symmetric tensors when their rank sum is below a specific threshold.

Findings

If the sum of real and complex ranks is at most 3 times the degree minus 1, the difference in decompositions is confined to a line, conic, or two disjoint lines.

The difference between the decompositions is fully determined by simple geometric objects under the given rank condition.

Abstract

We study the case of a real homogeneous polynomial $P$ whose minimal real and complex decompositions in terms of powers of linear forms are different. We prove that, if the sum of the complex and the real ranks of $P$ is at most $ 3\deg(P)-1$, then the difference of the two decompositions is completely determined either on a line or on a conic or two disjoint lines.