On Gröbner bases under specialization

We extend a result of Gianni and Kalkbrener concerning the stability of the Gröbner basis property under specializations. In particular, we prove that the Gröbner basis property is preserved if one specializes variables from bottom up in a Gröbner basis of a zero-dimensional radical ideal w.r.t. a lexicographical term order. The proof makes use of a connection between Gröbner bases and D-Gröbner bases which is proved separately. Computational aspects of this connection are discussed briefly.