Dimension-dependent bounds for Gröbner bases of polynomial ideals

Given a basis F of a polynomial ideal I in K[x1,…,xn] with degrees deg(F)≤d, the degrees of the reduced Gröbner 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)=d2Θ(n). This was established in Mayr and Meyer (1982), andDubé (1990).We modify both constructions in order to give worst case bounds depending on the ideal dimension proving that deg(G)=dnΘ(1)2Θ(r) for r-dimensional ideals (in the worst case).