Algebraic relations between solutions of Painlev\'e equations

We calculate model theoretic ranks of Painlev\'e equations in this article, showing in particular, that any equation in any of the Painlev\'e 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 Painlev\'e family is geometrically trivial, extending a result of Nagloo (2015). We also establish the orthogonality of various pairs of equations in the Painlev\'e families, showing at least generically, that all instances of nonorthogonality between equations in the same Painlev\'e family come from classically studied B{\"a}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 Painlev\'e equations. We give similar results for orthogonality relations between equations in different Painlev\'e families, and formulate some general questions which extend conjectures of Nagloo and Pillay (2011) on transcendence and algebraic independence of solutions to Painlev\'e equations. We also apply our analysis of ranks to establish some orthogonality results for pairs of Painlev\'e equations from different families. For instance, we answer several open questions of Nagloo (2016), and in the process answer a question of Boalch (2012).Comment: This manuscript replaces and greatly expands a portion of arXiv:1608.0475

Order one differential equations on nonisotrivial algebraic curves

In this paper we provide new examples of geometrically trivial strongly minimal differential algebraic varieties living on nonisotrivial curves over differentially closed fields of characteristic zero. These are systems whose solutions only have binary algebraic relations between them. Our technique involves developing a theory of $\tau$-forms, and building connections to deformation theory. This builds on previous work of Buium and Rosen. In our development, we answer several open questions posed by Rosen and Hrushovski-Itai