Isomonodromic deformations, Liouville form, \(\tau \) function.
The author makes comments and gives improvements of his earlier results from himself [Sur les déformations isomonodromiques I: singularités régulières. Mathematique of physique Séminaire ENS, Paris 1979-1982, Prog. Math. 37, 401-426 (1983; Zbl 0528.32017)]. Consider the family of Fuchsian systems \(dX/dt=(\sum _{j=1}^pA_j/(t-a_j))X\) in which the poles \(a_j\) are parameters. If the matrices-residua \(A_j\) depend on these parameters in such a way that the monodromy of the system doesn't change, then such a family is called isomonodromic. The isomonodromy condition (due to Schlesinger) is a completely integrable system whose solutions have the Painlevé property (to have only poles as movable singularities). The zeros of the logarithm of the \(\tau\) function locate the poles of these solutions. At such a pole the module of at least one entry of at least one of the matrices-residua \(A_j\) tends to infinity.
scientific article; zbMATH DE number 2162427
scientific article