Resolution of Singularities in Two Dimensions and the Stability of Integrals

Here j is a positive rational number and p = 0 or 1. Also using resolution of singularities, it can be shown that (j, p) is independent of U for small enough U . We refer to j as the growth index of S and to p as the multiplicity of this growth index. If S(x, y) is merely smooth, then it follows from [G1] that (1.2) still holds unless there is a smooth coordinate change fixing the origin after which the bisectrix intersects the Newton polygon of S in the interior of its horizontal ray (see below for the relevant definitions.) It is also true that except in those exceptional situations, CU is independent of U for small enough U . In the exceptional situations, then we can at least say there is a j such that for small enough U , for some CU > 0 we have MS,U ( ) < CU j , while for j′ > j there is no C ′ U for which the estimate MS,U ( ) < C ′ U j′ holds. Thus for the smooth situation, we also have a natural definition of the growth index of S(x, y) and its multiplicity.