Zeros of the Jimbo, Miwa, Ueno tau function

We introduce a family of local deformations for meromorphic connections on P 1 in the neighborhood of a higher rank (simple) singularity. Following the scheme in Malgrange [16-18] we use these local models to prove that the zeros of the tau function , introduced by Jimbo, Miwa, and Ueno in their pioneering work on " Birkhoff " deformations at irregular singular points [12], occur at precisely those points in the deformation space at which a certain Birkhoff-Riemann-Hilbert problem fails to have a solution. §1 Introduction The Riemann-Hilbert problem and monodromy preserving deformations. Suppose that A(x) is an p × p matrix with entries that are rational functions of x on P 1. Linear differential equations dψ dx = A(x)ψ, (1.0) arise in many important applications and have been studied intensively for more than 100 years. The nature of the singularities in the coefficient matrix A(x) at its poles has a lot to do with the character of the solutions to (1.0). For example, in the neighborhood of a point a where A(x) has but a simple pole it is well known that the equation (1.0) has a fundamental solution, Ψ(x), with polynomially limited growth as x → a (by which we mean that solutions are dominated near x = a by c|x − a| −N where c and N are constants). Higher order poles in A(x) typically produce local fundamental solutions with more complicated exponential-polynomial growth near the pole. Fundamental solutions to the equation (1.0) are usually not single valued. If one has a fundamental solution Ψ(x) defined for x in some small ball not containing any of the poles, {a 1 , a 2 ,. .. , a n }, of A(x) then it is possible to analytically continue Ψ(x) along paths in P 1 which avoid these poles. The analytic continuation depends only on the homotopy class of the path, so that the resulting function, Ψ(x), is defined not on P 1 but for x in the simply connected covering