Hyperelliptic sigma solutions of polynomial Hamiltonian systems
This research constructs local analytic solutions for polynomial Hamiltonian systems defined by specific Hamiltonian structures. It achieves this by reducing the complex system to a single scalar equation on a hyperelliptic curve, inverting it using Abelian integrals represented by sigma functions, and then reconstructing individual modes. This provides an algebraic geometry framework for solving these systems on higher-genus curves.