1. Download the polynomial system of {6,21,4} and the check polynomial.
6-21-4_poly_13_txt.gz
6-21-4_chkpoly_txt.gz

2. Expand these files.
gzip -d 6-21-4_poly_13_txt.gz
gzip -d 6-21-4_chkpoly_txt.gz

3. Start Singular

4. Execute the following commands on Singular:
system("--min-time", "0.01");
system("--ticks-per-sec", 100);
timer = 1; // Display setting of cpu time
ring r=0,(g,f,e,d,c,b,a),(Wp(18,12,6,5,4,3,2)); // Setting the weights
< "./6-21-4_chkpoly_txt"; // load of the check polynomial h of {6,21,4}
< "./6-21-4_poly_13_txt"; // load of the polynomial system i of {6,21,4}
option(prot); // Display setting of the progress
ideal g=groebner(i); // Groebner basis computation
reduce(h,g); // Computation of normal form of h w.r.t g

5. We get 0 by the above command. This shows that h in g=Id(i).