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Algebraic relations between solutions of Painlev\'e equations. (arXiv:1710.03304v1 [math.LO])
来源于:arXiv
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 equat 查看全文>>