Circular pentagons and real solutions of Painleve VI equations

TL;DR

This paper explores real solutions of Painleve VI equations by associating them with geometric circular pentagons, providing an algorithm to analyze their zeros, poles, and fixed points, and linking monodromy data to these geometric objects.

Contribution

It introduces a novel geometric approach using circular pentagons to study Painleve VI solutions and an algorithm to compute key solution features from these geometric objects.

Findings

Algorithm for counting zeros, poles, and fixed points of solutions.

Method to recover monodromy and parameters from geometric data.

Establishment of a geometric correspondence between solutions and circular pentagons.

Abstract

We study real solutions of a class of Painleve VI equations. To each such solution we associate a geometric object, a one-parametric family of circular pentagons. We describe an algorithm which permits to compute the numbers of zeros, poles, 1-points and fixed points of the solution on the interval x>1 and their mutual position. The monodromy of the associated linear equation and parameters of the Painleve VI equation are easily recovered from the family of pentagons.