# zbMATH — the first resource for mathematics

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
##### Keywords:
monodromy preserving deformation theory; poles
Full Text:
##### References:
 [1] Jimbo, M., Miwa, T. and Ueno, K., Physica, 2D (1981), 306. [2] Miwa, T., Publ RIMS, Kyoto Univ., 17 (1981), 665. [3] Ablowitz, M. J., Ramani, A. and Segur, H., Lett. Nuovo Cimento, 23 (1978), 333. [4] Okamoto, K., Proc. Japan Acad., 56A (1980), 264, 367. [5] Jimbo, M. and Miwa, T., Monodromy preserving deformation of linear ordinary differential equations with rational coefficients, II, Physica, 2D (1981), 407. · Zbl 1194.34166 [6] 1. [7] Oishi, S., /. Phys. Soc. Japan, 49 (1980), 1647. [8] Hilbert, D., Verh. des 3. internat. Math. Kongr. Heidelberg, (1904), 233. [9] Saito, T., Sugaku, 12 (1960), 145 (in Japanese. [10] Plemelj, J., Problems in the sense of Riemann and Klein, Interscience, 1960. [10] Birkhoff, G. D., Proc. Amer. Acad., 49 (1913), 521. [11] Sato, M., Miwa, T. and Jimbo, M., Publ. RIMS, Kyoto Univ., 14 (1978), 223; 15 (1979), 201, 577, 871; 16 (1980), 531. [12] Ueno, K., RIMS, Kokyuroku, 388 (1980), 102 (in Japanese). [14] Wu, T. T., McCoy, B. M., Tracy, C. A. and Barouch, E., Phys. Rev., B13(1976), 316. [13] Sato, M., Miwa, T. and Jimbo, M., Proc. Japan Acad., 53A (1977), 147, 153, 183. [14] Jimbo, M., Miwa, T., Mori, Y. and Sato, M., Physica, ID (1980), 80. [15] Jimbo, M. and Miwa, T., Proc. Japan Acad., 56A (1980), 143, 149, 269. Added in Proof’. By a letter from Professor B. Malgrange I learned that he also proved the Painleve property for the Schlesinger equation.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.