Title: Algebraic relations between solutions of Painlevé equations,
Abstract: 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 B{ä}cklund transformations. For instance, we show that if
at least one of $\alpha, \beta$ is transcendental, then $P_{II} (\alpha)$ is
nonorthogonal to $P_{II} ( \beta )$ if and only if $\alpha+ \beta \in \mathbb
Z$ or $\alpha - \beta \in \mathbb Z$. 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).