English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We study a Hamiltonian system without the Painlev\'e property and show thatit admits a kind of regularisation on a bundle of rational surfaces withcertain divisors removed, generalising Okamoto's spaces of initial conditionsfor the Painlev\'e differential equations. The system in question was obtainedby Takasaki as part of the Painlev\'e-Calogero correspondence and possesses thealgebro-Painlev\'e property, being related by an algebraic transformation tothe fourth Painlev\'e equation. We provide an atlas for the bundle of surfacesin which the system has a global Hamiltonian structure, with all Hamiltonianfunctions being polynomial in coordinates just as in the case of Okamoto'sspaces. We compare the surface associated with the Takasaki system with that ofthe fourth Painlev\'e equation, showing that they are related by a combinationof blowdowns and a branched double cover, under which we lift the birationalB\"acklund transformation symmetries of the fourth Painlev\'e equation toalgebraic ones of the Takasaki system, including a discrete Painlev\'eequation. We also discuss and provide more examples in support of the idea thatthere is a connection between the algebro-Painlev\'e property and similarnotions of regularisability, in an analogous way to how regular initial valueproblems for the Painlev\'e equations everywhere on Okamoto's spaces arerelated to the Painlev\'e property.

Takasaki's rational fourth Painlev\'e-Calogero system and geometric regularisability of algebro-Painlev\'e equations