Superintegrable systems and Riemann-Roch theorem

TL;DR

This paper explores how algebraic geometry's reduction algorithm, based on the Riemann-Roch theorem, can be applied to construct finite-dimensional superintegrable systems by linking reduced divisors to integrals of motion.

Contribution

It introduces a novel application of the Riemann-Roch theorem's reduction algorithm to the construction of superintegrable systems, connecting algebraic geometry with integrable systems.

Findings

Reduction algorithm yields unique reduced divisors

Coordinates of reduced divisors serve as integrals of motion

Method enables systematic construction of superintegrable systems

Abstract

In algebraic geometry, there is a reduction algorithm that transforms the unreduced divisor into a unique reduced divisor, which existence is guaranteed by the Riemann-Roch theorem. We discuss application of this algorithm to construction of finite-dimensional superintegrable systems with $n$ degrees of freedom identifying coordinates of the reduced divisor with integrals of motion.