On tensors that are determined by their singular tuples

TL;DR

This paper investigates the set of singular tuples of complex multisymmetric tensors, establishing conditions under which tensors are uniquely determined by these tuples or form a one-dimensional family.

Contribution

It characterizes the tensors determined by their singular tuples based on degree conditions and the existence of the hyperdeterminant, revealing uniqueness or dimensionality of the fiber.

Findings

Tensors with at least one odd degree component are projectively unique.

If all degrees are even, the set of tensors sharing the same singular tuples forms a 1-dimensional space.

The existence of the hyperdeterminant is linked to the triangular inequality condition.

Abstract

In this paper we study the locus of singular tuples of a complex valued multisymmetric tensor. The main problem that we focus on is: given the set of singular tuples of some general tensor, which are all the tensors that admit those same singular tuples. Assume that the triangular inequality holds, that is exactly the condition such that the dual variety to the Segre-Veronese variety is an hypersurface, or equivalently, the hyperdeterminant exists. We show in such case that, when at least one component has degree odd, this tensor is projectively unique. On the other hand, if all the degrees are even, the fiber is an $1$-dimensional space.