On Dh-Closed Gr(?)bner Bases

This paper introduces the notions of a dh-closed homogeneous Grobner basis and a dh-closed graded ideal for the polynomial ring R[t] over an N-graded K-algebra R with a Grobner basis theory, respectively for the free K-algebra K(X1,…,Xn,T), and demonstrates a general algorithmic principle of obtaining a Grobner basis for an idealⅠgenerated by non-homogeneous elements and thereby obtainting a homogeneous Grobner basis for the homogenization ideal ofⅠ, by passing to dealing with the homogenized generators; Secondly, all homogeneous Grobner basis in R[t] that correspond bijectively to all Grobner basis in R, respectively all homoge-neous Grobner basis in K(X,T) that correspond bijectively to all Grobner basis in K(X), are characterized; Finally, all graded ideals in R[t] that correspond bijectively to all ideals in R, respectively all graded ideals in K(X,T) that correspond bijectively to all ideals in K(X), are characterized in terms of Grobner basis.