From the paper of Encinas and Hauser:
The coefficient ideal of an ideal of at a with respect to is an ideal in which is built from the coefficients of the Taylor expansion of the elements of with respect to the equations defining . Let and be regular systems of parameters of and so that defines in . For in denote by the elements of so that holds after passage to the completion. Then we set
Let us choose as the hypersurface. Then:
Let be a closed subscheme of a regular scheme . We want to resolve the singularity of . If there exists an regular hypersurface such that , then we may replace by . So we may (have to) assume that is not contained in such hypersurfaces. Instead, we have for each point a ``hypersurface of maximal contact'' . is not canonical, but is good enough to define the invariant and then (afterwards) determine the center of blow up.