On Gröbner bases under specialization (Q1311607)

Let \(K\) be a field, \(K[Y,X] = K[Y_ 1, \dots, Y_ m, X_ 1, \dots, X_ n]\) the polynomial ring over \(K\), and let \(<\) be a lexicographical term order on the set of power-products in \(K[Y,X]\) such that \(Y_ i<X_ j\), for all \(i,j\). Let \(G\) be a Gröbner basis of an ideal \(I\) in \(K[Y,X]\), \(z = (z_ 1, \dots, z_ m) \in \overline K^ m\) a zero of \(I \cap K[Y]\), and \(G(z)\) (resp. \(I(z))\) the set of polynomials in \(\overline K[X]\) obtained by substituting \(z\) for \(Y\) in all elements of \(G\) (resp. \(I)\). The author's main result is: If \(I \cap K[Y]\) is zero-dimensional and radical, then \(G(z)\) is a Gröbner basis of \(I(z)\).