The Computational Complexity of the Resolution of Plane Curve Singularities

We present an algorithm which computes the resolution of a plane curve singularity at the origin defined by a power series with coefficients in a (not necessarily algebraically closed) field k of characteristic zero. We estimate the number of £-operations necessary to compute the resolution and the conductor ideal of the singularity. We show that the number of A:-operations is polynomial^ bounded by the complexity of the singularity, as measured for example by the index of its conductor ideal. Our algorithm involves calculations over reduced rings with zero divisors, and employs methods of deformation theory to reduce the consideration of power series to the consideration of polynomials. The problem of resolving singularities is of fundamental interest in modern algebraic geometry. In this paper we make a small step toward approaching this problem from the point of view of computational complexity. We present an algorithm, suitable for machine implementation, which computes the resolution of a plane curve singularity—that is, a singularity at the origin defined by a formal power series F in two variables x and y over a field k . As we describe it, the algorithm requires that k be of characteristic zero (or at least of "large" characteristic) but this hypothesis can certainly be removed at the expense of some complications. The algorithm obtains explicit equations for the blowingup of the singularity, and therefore yields all of the interesting invariants of the singularity, such as its conductor and its Milnor number. We also provide upper bounds for the number of /c-operations needed for the operation of the algorithm. The problems we consider in this paper have a long history. In [18], Kung and Traub consider the complexity of Newton's method for solving analytic equations. There, they present estimates for the number of times Newton's method must be applied to obtain an approximate solution to an analytic equation before an iterative method can be employed to refine the solution. This process is closely related to the resolution problem. Berry, in [3], considers the complexity of Coates' algorithm [8] for computing Puiseux expansions. Chudnovsky and Chudnovsky [7] have looked at computing Puiseux expansions from the point of view of differential equations. The work of Duval and Dicrescenzo Received March 14, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 14B05, 68Q40. This research was supported in part by a Rackham Postdoctoral Research Fellowship at the University of Michigan and by an NSF Postdoctoral Fellowship. ©1990 American Mathematical Society 0025-5718/90 $1.00+ $.25 per page