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:'''Output''' A [[Gröbner basis]] ''G'' for {{mvar|I}}
:# ''G'' := ''F''
:# For every ''f<sub>i</sub>'', ''f<sub>j</sub>'' in ''G'', denote by ''g<sub>i</sub>'' the leading term of ''f<sub>i</sub>'' with respect to the given [[monomial ordering]], and by ''a<sub>ij</sub>'' the [[least common multiple]] of ''g<sub>i</sub>'' and ''g<sub>j</sub>''.
:# Choose two polynomials in ''G'' and let {{math|1=''S''<sub>''ij''</sub> = {{sfrac|''a''<sub>''ij''</sub> | ''g''<sub>{{var|i}}</sub>}} ''f''<sub>{{var|i}}</sub> − {{sfrac|''a''<sub>''ij''</sub> | ''g''<sub>''j''</sub>}} ''f''<sub>''j''</sub>}} ''(Note that the leading terms here will cancel by construction)''.
:# Reduce ''S''<sub>''ij''</sub>, with the [[multivariate division algorithm]] relative to the set ''G'' until the result is not further reducible. If the result is non-zero, add it to ''G''.
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