Nonlinear Kalman varieties

TL;DR

This paper explores nonlinear Kalman varieties, extending the classical linear case to arbitrary projective varieties, and investigates their geometric invariants and determinantal equations, with applications in quantum chemistry and optimization.

Contribution

It generalizes the known linear Kalman variety results to nonlinear cases, including hypersurfaces, and provides new determinantal descriptions of their defining equations.

Findings

Determined dimensions, degrees, and singularities of nonlinear Kalman varieties.

Extended determinantal equations to hypersurface cases.

Connected geometric invariants to applications in quantum chemistry and optimization.

Abstract

We study the locus of square matrices having at least one eigenvector on a prescribed algebraic variety $X$. When $X$ is a linear subspace, this data locus is known as the Kalman variety of $X$ and was studied first by Ottaviani and Sturmfels. Motivated by recent applications to quantum chemistry and optimization, in this work, we focus on nonlinear Kalman varieties, that is, Kalman varieties relative to arbitrary projective varieties $X$. We study the basic invariants of these varieties, such as their dimensions, degrees, and singularities. Furthermore, Ottaviani and Sturmfels provide determinantal equations in the linear case. We generalize their result to Kalman varieties of hypersurfaces by providing a determinantal-like description of their equation.