Submission date: 9 October 2017

Abstract:

This manuscript replaces and greatly expands a portion of arXiv:1608.04756
We calculate model theoretic ranks of Painlevé equations in this article, showing in particular, that any equation in any of the Painlevé families has Morley rank one, extending results of Nagloo and Pillay (2011). We show that the type of the generic solution of any equation in the second Painlevé family is geometrically trivial, extending a result of Nagloo (2015).
We also establish the orthogonality of various pairs of equations in the Painlevé families, showing at least generically, that all instances of nonorthogonality between equations in the same Painlevé family come from classically studied Bäcklund transformations. For instance, we show that if at least one of α, β is transcendental, then P_{II} (α) is nonorthogonal to P_{II} ( β ) if and only if α+ β in Z or α - β \in Z. Our results have concrete interpretations in terms of characterizing the algebraic relations between solutions of Painlevé equations. We give similar results for orthogonality relations between equations in different Painlevé families, and formulate some general questions which extend conjectures of Nagloo and Pillay (2011) on transcendence and algebraic independence of solutions to Painlevé equations. We also apply our analysis of ranks to establish some orthogonality results for pairs of Painlevé equations from different families. For instance, we answer several open questions of Nagloo (2016), and in the process answer a question of Boalch (2012).

Mathematics Subject Classification: 34M55, 03C60

Keywords and phrases:

Full text arXiv 1710.03304: pdf, ps.