Reduction of Singularities of the Differential Equation Ady = Bdx

Let k [ [X, Y]] be the ring of formal power series in two letters over an algebraically closed base field k of characteristic zero, let A, B C k [ [X, Y]] be elements without non-unit common divisor, and consider the differential equation A (x, y) dy B (x, y) dx. By a solution (at the origin) of the equation we mean an analytic branch (centered at the origin) which satisfies the equation; such a branch is represented by a pair of formal power series x cit + c2t2? + * * , y d,t + d2t2 + * , with x, y not both =0, such that A(x,y) (Vic.ti-1) -B(x,y) (Ejdjtj-1). Let r=min{subdA,subdB}. The number r is a measure of the complexity of the equation at the origin, and in particular the origin is called singular for Ady = Bdx if r ? 1. In Theorem 12 below we show how, after a finite number of transformations of the form X' = X, Y'Y/X, translations tacitly included, the solutions of Ady -Bdx can be made to go over into solutions of equations with r ? 1. In Theorems 8 and 10 the case r = 1 is analyzed; and we thus have, in one direction at any rate, a complete theory. In Theorem 13 we show that the set S of solutions can be divided into a finite number of subsets Si, S = U Si, such that any pair in a given Si are equisingular in the sense of [1]. The other theorems are for the most part preparatory. Undoubtedly some of them (in particular Theorems 2, 5, 7, 14) are known, but they are included for the sake of coherence.