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:'''Output''' A [[Gröbner basis]] for ''I''
:# ''G'' := ''F''
:# For every ''f<sub>i</sub>'', ''f<sub>j</sub>'' in ''G'', denote by ''g<sub>i</sub>''
:# Choose two polynomials in ''G'' and let ''S''<sub>''ij''</sub> = (''a''<sub>''ij''</sub> / ''g''<sub>''i''</sub>) ''f''<sub>''i''</sub> − (''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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