Iterative polynomial regularisation of PIV-type: Hamiltonian systems and Newton polygons

TL;DR

The paper addresses the problem of identifying minimal polynomial Hamiltonians for PIV-type and quasi-PIV systems by applying iterative polynomial regularisation and analyzing the resulting Newton polygons. By examining a generalised Okamoto Hamiltonian for $P_{ ext{IV}}$, a modified version, and the mixed quasi-PIV case, it shows that minimal representatives correspond to Newton polygons of area $2$ (diamond-like) and, in some instances, require one or two regularisation steps to minimize the highest total degree. In the quasi-PIV mixed case, a reduction from area $5$ to $\frac{7}{2}$ is observed, suggesting a more economical Hamiltonian form; however, it remains open whether even smaller polygons exist for other Hamiltonians. The work provides a geometric/combinatorial criterion for classifying Painlevé-type Hamiltonians and offers insights toward the quasi-Painlevé equivalence problem and related reductions in integrable systems and orthogonal-polynomial models.

Abstract

We study several polynomial Hamiltonian systems of PIV-type (including the mixed case quasi-PIV), 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.