Eigenschemes of Ternary Tensors

TL;DR

This paper investigates the geometric structure of eigenschemes of ternary tensors, providing descriptions, characterizations, and algorithms for their analysis using algebraic geometry techniques.

Contribution

It offers a birational description of eigenschemes of general ternary symmetric tensors, characterizes their defining polynomials, and develops algorithms for tensor reconstruction.

Findings

Dimension of the eigenscheme variety computed

Algorithms for checking and reconstructing eigenschemes developed

Geometric characterization of reduced zero-dimensional eigenschemes provided

Abstract

We study projective schemes arising from eigenvectors of tensors, called eigenschemes. After some general results, we give a birational description of the variety parametrizing eigenschemes of general ternary symmetric tensors and we compute its dimension. Moreover, we characterize the locus of triples of homogeneous polynomials defining the eigenscheme of a ternary symmetric tensor. Our results allow us to implement algorithms to check whether a given set of points is the eigenscheme of a symmetric tensor, and to reconstruct the tensor. Finally, we give a geometric characterization of all reduced zero-dimensional eigenschemes. The techniques we use rely both on classical and modern complex projective algebraic geometry.