Iterative polynomial regularisation of $\text{P}_{\text{IV}}$-type: Hamiltonian systems and Newton polygons

Abstract: We study several polynomial Hamiltonian systems of $\text{P}{\text{IV}}$-type (including the mixed case quasi-$\text{P}{\text{IV}}$), and show that via the iterative process of polynomial regularisation, it is possible to identify the "minimal" Hamiltonian system. The selected Hamiltonian function is associated with the Newton polygon with minimal area and smallest highest total degree.