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Painlevé property of monodromy preserving deformation equations and the analyticity of \(\tau\) functions. (English) Zbl 0605.34005
The author aims to prove the two conjectures in monodromy preserving deformation theory proposed earlier by himself [ibid. 17, 665-686 (1981; Zbl 0505.35070)]. The first of them states for the general monodromy preserving deformation equations the property possessed by the six Painlevé equations: The singularities of solutions to the monodromy preserving deformation equations are poles except for the fixed singularities. The second deals with the \(\tau\)-function, which was introduced in an article by M. Jimbo, the author and K. Ueno [Physica D 2, 306-352 (1981)], and formulated as follows: The \(\tau\)- function is holomorphic except at the fixed singularities. To prove these statements the author uses quantum field theory techniques.

MSC:
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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[1] Jimbo, M., Miwa, T. and Ueno, K., Physica, 2D (1981), 306.
[2] Miwa, T., Publ RIMS, Kyoto Univ., 17 (1981), 665.
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