In an upcoming exam we have to do Gröbnber-Basis computation with Buchberger's algorithm. A typical example looks like this:
$$ \langle f_1,f_2 \rangle $$
Then I compute the S-Polynomial $S(f_1,f_2)$. Most of the time $S(f_1,f_2)$ is an ugly expression so I use linear combinations of $\langle f_1, f_2, S(f_1,f_2)\rangle$ and obtain a new representation of the ideal with nice polynomials $\langle f_1',f_2',f_3' \rangle$.
Now, I have to restart Buchberger's algorithm because I have changed the ideal representation and compute $S(f_1',f_2'), S(f_1',f_3') \dots$. Usually $\langle f_1',f_2',f_3' \rangle$ is a valid Gröber basis already. But to compute the three (or more) S-Polynomials takes a lot of time. Is there are a (fast computable) criteria to check whether a given basis is already a Gröbner-Basis and to abort the computation?
