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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.
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Differential Equations for Approximate Solutions of Painlevé Equations: Application to the Algebraic Solutions of the Painlevé-III $({\rm D}_7)$ Equation
It is well known that the Painlevé equations can formally degenerate to autonomous differential equations with elliptic function solutions in suitable scaling limits. A way to make this degeneration rigorous is to apply Deift-Zhou steepest-descent techniques to a Riemann-Hilbert representation of a family of solutions. This method leads to an explicit approximation formula in terms of theta functions and related algebro-geometric ingredients that is difficult to directly link to the expected limiting differential equation. However, the approximation arises from an outer parametrix that satisfies relatively simple conditions. By applying a method that we learned from Alexander Its, it is possible to use these simple conditions to directly obtain the limiting differential equation, bypassing the details of the algebro-geometric solution of the outer parametrix problem. In this paper, we illustrate the use of this method to relate an approximation of the algebraic solutions of the Painlevé-III (D$$_7$$) equation valid in the part of the complex plane where the poles and zeros of the solutions asymptotically reside to a form of the Weierstraß equation.
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- PAR ID:
- 10509346
- Publisher / Repository:
- SIGMA
- Date Published:
- Journal Name:
- Symmetry, Integrability and Geometry: Methods and Applications
- ISSN:
- 1815-0659
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3842/SIGMA.2024.008
