This question could refer to any computer algebra system which has the ability to compute the Groebner Basis from a set of polynomials (Mathematica, Singular, GAP, Macaulay2, MatLab, etc.).
I am working with an overdetermined system of polynomials for which the full groebner basis is too difficult to compute, however it would be valuable for me to be able to print out the groebner basis elements as they are found so that I may know if a particular polynomial is in the groebner basis. Is there any way to do this?
If you implement Buchberger's algorithm on your own, then you can simply print out the elements as the are found.
If you have Mathematica, you can use this code as your starting point.
https://www.msu.edu/course/mth/496/snapshot.afs/groebner.m
See the function BuchbergerSteps.
Due to the way the Buchberger algorithm works (see, for instance, Wikipedia or IVA), the partial results that you could obtain by printing intermediate results are not guaranteed to constitute a Gröbner basis.
Depending on your ultimate goal, you may want to try instead an algorithm for triangularization of ideals, such as Ritt-Wu's algorithm (see IVA or Shang-Ching Chou's book). This is somewhat similar to reduction to row echelon form in Linear Algebra, and you may interrupt the algorithm at any point to get a partially reduced system of polynomial equations.
来源：https://stackoverflow.com/questions/9506705/return-elements-of-the-groebner-basis-as-they-are-found