We propose an improvement upon the standard algorithm for computing the kernel of a polynomial map, assuming that the map sends monomials into monomials. Rather than computing a Gröbner basis in the joint polynomial ring, and then selecting only the elements of interest, we show that a moderate number of iterations of the Buchberger algorithm in the variables of the domain suffices.
"An algorithm to calculate the kernel of certain polynomial ring homomorphisms." Experiment. Math. 4 (3) 227 - 234, 1995.