Nonlinear Kalman varieties
Flavio Salizzoni, Luca Sodomaco, Julian Weigert
TL;DR
This paper generalizes Kalman varieties to nonlinear Kalman varieties ${\mathcal K}(X)$ for arbitrary projective varieties $X$, studying their dimensions, degrees, and singularities, and connects them to determinantal descriptions. A key construction is the Kalman matrix of order $d$, $K_d(C_X)$, built via Veronese embeddings and the $d$-th symmetric power representation, which yields a determinantal description that contains ${\mathcal K}(X)$; in the hypersurface case, the determinant factorizes into discriminant and generalized Kalman factors $p_{\mu}$ corresponding to ${\mathcal K}_{\mu}(f)$, with explicit degree formulas. The authors provide a detailed factorization pattern, transversality arguments, and a precise degree formula for generalized Kalman varieties ${\mathcal K}_{\mu}(f)$, together with concrete examples (e.g. plane conics) to illustrate the framework. The study of singular loci reveals a structured decomposition of Sing$({\mathcal K}(X))$ into contributions from the components of $X$ and its singularities, with degeneration to unions of hyperplanes enabling degree computations and showing containment in minors of the Kalman matrix. Overall, the work extends algebraic-geometric control-theory concepts to nonlinear settings and offers both theoretical invariants and computational tools relevant to quantum chemistry and constrained 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.