Castelnuovo-Mumford regularity and extended degree
  The main result of this paper shows that the Castelnuovo-Mumford regularity of the tangent cone of a local ring is effectively bounded by the dimension and any extended degree. From this it follows that there are only a finite number of Hilbert-Samuel functions of local rings with given dimension and extended degree. 
Maria Evelina Rossi, Ngo Viet Trung, Giuseppe Valla