# On the ranks of the third secant variety of Segre-Veronese embeddings

## TL;DR

This paper establishes bounds on the tensor rank and border rank for partially symmetric tensors, particularly in the Segre case, revealing the range of possible ranks under various conditions.

## Contribution

It provides new upper bounds for the rank of border rank 3 tensors and characterizes the spectrum of attainable ranks in the Segre case.

## Key findings

- All ranks from 3 to k-1 occur for border rank 3 tensors.
- If all dimensions are at least 3, ranks between k and 2k-1 also occur.
- The results give a comprehensive view of rank possibilities for these tensors.

## Abstract

We give an upper bound for the rank of the border rank 3 partially symmetric tensors. In the special case of border rank 3 tensors $T\in V_1\otimes \cdots \otimes V_k$ (Segre case) we can show that all ranks among 3 and $k-1$ arise and if $\dim V_i \geq 3$ for all $i$'s, then also all the ranks between $k$ and $2k-1$ arise.