Degrees of Kalman varieties of tensors

TL;DR

This paper studies the algebraic geometry of Kalman varieties of tensors, extending previous work to partially symmetric tensors, and provides formulas and asymptotic analysis for their degrees, highlighting the role of isotropic vectors.

Contribution

It extends the theory of Kalman varieties to partially symmetric tensors and introduces a generating function for their degrees with asymptotic analysis.

Findings

Derived a generating function for degrees of Kalman varieties.

Analyzed asymptotics of the degrees using analytic methods.

Identified the significance of isotropic vectors in tensor spectral theory.

Abstract

Kalman varieties of tensors are algebraic varieties consisting of tensors whose singular vector $k$-tuples lay on prescribed subvarieties. They were first studied by Ottaviani and Sturmfels in the context of matrices. We extend recent results of Ottaviani and the first author to the partially symmetric setting. We describe a generating function whose coefficients are the degrees of these varieties and we analyze its asymptotics, providing analytic results \`a la Zeilberger and Pantone. We emphasize the special role of isotropic vectors in the spectral theory of tensors and describe the totally isotropic Kalman variety as a dual variety.