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.
Fausto Di Biase. Rüdiger Urbanke. "An algorithm to calculate the kernel of certain polynomial ring homomorphisms." Experiment. Math. 4 (3) 227 - 234, 1995.