Some integrable systems of algebraic origin and separation of variables

TL;DR

This paper explores integrable systems derived from algebraic curves, demonstrating how separation of variables leads to Poisson commuting coefficients, with applications to interpolation polynomials and elliptic curves.

Contribution

It generalizes previous results by formulating relations that ensure Poisson commutativity in algebraic integrable systems using separation of variables.

Findings

Poisson commute of coefficients of algebraic curves

Relations implying Poisson commutativity established

Examples include interpolation polynomials and elliptic curves

Abstract

A plane algebraic curve whose Newton polygone contains d lattice points can be given by d points it passes through. Then the coefficients of its equation Poisson commute having been regarded as functions of coordinates of those points. It is observed in the work by O.Babelon and M.Talon, 2002. We formulate a generalization of this fact in terms of separation of variables and prove relations implying the Poisson commutativity. The examples of the integrable systems obtained this way include coefficients of the Lagrange and Hermit interpolation polynomials, coefficients of the Weierstrass models of curves.