Dimension-dependent bounds for Gröbner bases of polynomial ideals

Given a basis F of a polynomial ideal I in K[x"1,...,x"n] with degrees deg(F)@?d, the degrees of the reduced Grobner basis G w.r.t. any admissible monomial ordering are known to be double exponential in the number of indeterminates in the worst case, i.e. deg(G)=d^2^^^@Q^^^(^^^n^^^). This was established in Mayr and Meyer (1982) andDube (1990). We modify both constructions in order to give worst case bounds depending on the ideal dimension proving that deg(G)=d^n^^^@Q^^^(^^^1^^^)^2^^^@Q^^^(^^^r^^^) for r-dimensional ideals (in the worst case).