rank reduction of a class of pfaffian systems in two variables
Several algorithms exist to reduce the rank of an ordinary linear differential system at a point, say 0, to its minimal value, the Poincar rank (also, sometimes called true Poincar rank). We extend Levelt algorithm, based on the existence of stationary sequences of free lattices, to completely integrable Pfaffian systems with normal crossings in two variables dY = (1/x p+1 A(x, y)dx + 1/ y q+1 B(x, y)dy)Y where A, B are mm matrices with entries in C[[x, y]] and p, q are non negative integers. The algorithm returns a completely integrable Pfaffian system with normal crossings dZ = (1/ x p+1 A(x, y)dx + 1/ y q+1 B(x, y)dy)Z equivalent to the initial one through a formal meromorphic gauge transformation at the origin 0, the integers p, q being simultaneously and individually the smallest possible. We, thus, set up a first step towards the explicit calculation of formal solutions of such systems.The particular case of a regular singular point at 0 is equivalent to p = q = 0, a condition easily checked by applying the algorithm.
rank pfaffian systems levelt algorithm poincar&#233