In my talk I will consider affine hypersurface singularities X defined over an algebraically closed field of characteristic zero. In order to get a deeper understanding of X it is useful to construct an overweight deformation to a better behaved variety. In our setting the latter means one that is given by a binomial prime ideal. By discussing a concrete example, I will illustrate how this can be attained if X is of dimension one. Within this, I will introduce the notion of a weighted characteristic polyhedron. Then I will discuss a generalization for arbitrary dimensional X and state a precise characterization when the construction succeeds to provide an overweight deformation. In the remaining time I will explain the idea for the proof. This is joint work with Hussein Mourtada.