Inventiones math. 64, 417-429 (1981) In vs~l tlons$ mathematicae 9 Springer-Verlag 1981 Estimation of Lojasiewicz Exponents and Newton Polygons Ben Lichtin* Institute of Advanced Study, Department of Mathematics, Princeton, NJ 08540, USA Section 1 Computation of topological invariants of an isolated singularity for a complex analytic map germ f: (C", 0) --, (C, 0), has been facilitated by the introduction of the Newton polygon F+(f) of f. For those germs f which are "non-degen- erate" with respect to their polygon, a modification of C" can be constructed by "dualizing the polygon." This allows one to place f in a normal crossing form in a neighborhood of any point in the preimage of the origin in C". If the invariant can be expressed when f is in this monomial form in terms of the ex- ponents of the monomials then the standard procedure is to interpret these ex- ponents in terms of the geometrical characteristics of the polygon. In this way, Varcenko has computed the zeta function of the monodromy operator [-9] and the oscillatory index [10] for a nondegenerate germ. One invariant as yet not computed is the "C o degree of sufficiency" v I for the map germ f. This number is the smallest integer v satisfying the property that terms of order greater than v do not modify the topological type of the polynomial consisting of those terms in f of order at most v. More precisely, let fv be the polynomial consisting of all terms in f with order at most v ("vth jet of f at 0"). Then for any germ g all of whose terms have order at least v +1, f~+g is topologically equivalent to fv at 0, as germs of maps on C". v I is the smallest integer with this property, fvs is a "C o sufficient jet." A natural de- sire is to express vy in terms of F+ (f). We do this for n=2 by imposing a nondegeneracy condition stronger than that imposed by Varcenko. Nonetheless, this is still a "generic" condition in the sense defined in Sect. (4). In Sect. (2), we present a few of the salient details concerning toroidal mo- difications, not all of which have appeared in print by now. Sect. (3) contains the derivation of our formula for the generic v I in the case of two variables. Before proceeding, however, we present, without very much comment, the formal constructions from analytic geometry which act as our computational guide. * Supported in part by NSF grant MCS 77-18723 A03