We point out two major differences to Hironaka's famous proof. One is the visible change of the singularity in each step. The second is that this new proof does not use any induction on
the dimension of the variety. Even if one takes the expansion for granted, the description
of resolution given above was not quite correct. The final proof will use another iteration of this procedure. After expanding one equation, one can extract some coefficients. These coefficients have to be expanded again, then the coefficients of the coefficients, etc. This leads to the notion of a web, which is not treated here.
After this rather crude description of a very complicated mechanism, we can indicate the content of the paper that follows. It contains the notation which is necessary to deal with the huge amount of information contained in the expansion. Then there is a proof for the existence of weighted initial forms in great generality, maybe more general than is needed for the purpose of
resolution. Finally the existence of an expansion as indicated above is proved.
; With foreword by U. Orbanz.