Equations of tensor eigenschemes

TL;DR

This paper investigates tensor eigenschemes using algebraic and geometric methods, providing determinantal equations and geometric conditions for their structure, especially in the symmetric case.

Contribution

It introduces a characterization of eigenschemes through linear equations and establishes a geometric necessary condition for 0-dimensional eigenschemes.

Findings

Determinantal equations of eigenschemes are characterized via linear equations.

A geometric necessary condition for 0-dimensional eigenschemes is established.

Results apply to both general and symmetric tensor cases.

Abstract

We study schemes of tensor eigenvectors from an algebraic and geometric viewpoint. We characterize determinantal defining equations of such eigenschemes via linear equations in their coefficients, both in the general and in the symmetric case. We give a geometric necessary condition for a 0-dimensional scheme to be an eigenscheme.