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Abstract In this manuscript, we make major progress classifying algebraic relations between solutions of Painlevé equations. Our main contribution is to establish the algebraic independence of solutions of various pairs of equations in the Painlevé families; for generic coefficients, we show that all algebraic relations between solutions of equations in the same Painlevé family come from classically studied Bäcklund transformations. We also apply our analysis of ranks to establish some transcendence results for pairs of Painlevé equations from different families. In that area, we answer several open questions of Nagloo, and in the process answer a question of Boalch. We calculate model‐theoretic ranks of all Painlevé equations in this article, extending results of Nagloo and Pillay. 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. We give the first model‐theoretic analysis of several special families of the third Painlevé equation, proving results analogous to Nagloo and Pillay. We also give a novel new proof of the irreducibility of the third, fifth, and sixth Painlevé equations using recent work of Freitag, Jaoui, and Moosa. Our proof is fundamentally different from the existing transcendence proofs by Watanabe, Cantat and Loray, or Casale and Weil. 

Abstract We study the structure of the solution sets in universal differential fields of certain differential equations of order two, the Poizat equations, which are particular cases of Liénard equations. We give a necessary and sufficient condition for strong minimality for equations in this class and a complete classification of the algebraic relations for solutions of strongly minimal Poizat equations. We also give an analysis of the non-strongly minimal cases as well as applications concerning the Liouvillian and Pfaffian solutions of some Liénard equations. 

We prove the Ax-Lindemann-Weierstrass theorem with derivatives for the uniformizing functions of genus zero Fuchsian groups of the first kind. Our proof relies on differential Galois theory, monodromy of linear differential equations, the study of algebraic and Liouvillian solutions, differential algebraic work of Nishioka towards the Painlevé irreducibility of certain Schwarzian equations, and considerable machinery from the model theory of differentially closed fields. Our techniques allow for certain generalizations of the Ax-Lindemann-Weierstrass theorem that have interesting consequences. In particular, we apply our results to give a complete proof of an assertion of Painlevé (1895). We also answer certain cases of the André-Pink conjecture, namely, in the case of orbits of commensurators of Fuchsian groups.