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:# 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''.
:# Repeat steps
:# Output ''G''
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