The totally nonnegative part of the finite Toda lattice via a reducible rational curve

TL;DR

This paper characterizes the totally nonnegative part of the finite Toda lattice using algebraic geometry, specifically through the real part of the generalized Jacobi variety of a reducible rational curve, revealing its structure as a semi-algebraic set.

Contribution

It provides an algebro-geometric description of the totally nonnegative finite Toda lattice via reducible rational curves and their Jacobi varieties, establishing an isomorphism with a connected component of the real Jacobi variety.

Findings

The totally nonnegative part of the Toda lattice corresponds to a connected component of the real Jacobi variety.

The Toda flow induces a linear flow on the generalized Jacobi variety.

The structure of the totally nonnegative part is semi-algebraic and explicitly characterized.

Abstract

A totally nonnegative matrix is a real-valued matrix whose minors are all nonnegative. In this paper, we concern with the totally nonnegative structure of the finite Toda lattice, a classical integrable system, which is expressed as a differential equation of square matrices. The Toda flow naturally translates into a (multiplicative) linear flow on the (generalized) Jacobi variety associated with some reducible rational curve $X$. This correspondence provides an algebro-geometric characterization of the totally positive part of the Toda equation. We prove that the totally nonnegative part of the finite Toda lattice is isomorphic to a connected component of $\mathrm{Jac}(X)_{\mathbb{R}}$, the real part of the generalized Jacobi variety $\mathrm{Jac}(X)$, as semi-algebraic varieties.