  We improve certain degree bounds for Grobner bases of polynomial ideals in
generic position. We work exclusively in deterministically verifiable and
achievable generic positions of a combinatorial nature, namely either strongly
stable position or quasi stable position. Furthermore, we exhibit new
dimension- (and depth-)dependent upper bounds for the Castelnuovo-Mumford
regularity and the degrees of the elements of the reduced Grobner basis (w.r.t.
the degree reverse lexicographical ordering) of a homogeneous ideal in these
positions.
