Stratified Canonical Forms of Matrix Valued Functions in a Neighborhood of a Transition Point

1.1. Let X be a complex analytic manifold or an algebraic variety over a perfect field k of dimension d, let xo be a point in X(k), and let g: X→ gl(n, k) be a germ of a matrix valued analytic (respectively, polynomial) function on X in an analytic (respectively, Zariski) neighborhood of xo. Questions regarding the possibility of reduction of such functions to the Jordan form j(x), or to some other normal form in a neighborhood of a point xo ofX,by conjugating it by a matrix function u: X → GL(n, k) in the same category; and questions about the existence of a conjugating matrix function u with suitable analytic properties, play an important role in various branches of analysis, geometry, and mathematical physics. In particular, such questions arise frequently and unavoidably in nearly all the main branches of the theory of ordinary ([Gn], [W1]–[W3]) and partial ([CH], [GV], [HYP], [N], [P1]–[P3]) differential equations, dynamical systems and their bifurcations ([CLW]), the perturbation theory of linear operators ([B], [K]), and the quantum field theory ([BT]). A reduction to canonical forms was used especially widely, systematically, and successfully in the general theory of hyperbolic systems of PDE since the pathbreaking works of Petrowsky ([P1]–[P3]), at least. Indeed, many of the principal general results on the existence and uniqueness of the solutions of the Cauchy (initial value) and mixed (initial-boundary value) problems, on the correctness of these problems, on explicit constructions of the fundamental and other solutions or parametrices for these problems, and on properties and asymptotics of their solutions, etc., for hyperbolic systems, have been obtained using a reduction of the principal symbols of these systems to suitable