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If ''G''<sub>prev</sub> is an already computed Gröbner basis (''f''<sub>2</sub>, …, ''f''<sub>''m''</sub>) and we want to compute a Gröbner basis of (''f''<sub>1</sub>) + ''G''<sub>prev</sub> then we will construct matrices whose rows are ''m'' ''f''<sub>1</sub> such that ''m'' is a monomial not divisible by the leading term of an element of ''G''<sub>prev</sub>.
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This strategy allows the algorithm to apply two new criteria based on what Faugère calls ''signatures'' of polynomials. Thanks to these criteria, the algorithm can compute Gröbner bases for a large class of interesting polynomial systems, called ''[[regular
== Implementations ==
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