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:# Add all the nonzero polynomials resulting from step 3 to ''F'', and repeat steps 1-4 until nothing new is added.
The polynomial ''S''<sub>''ij''</sub> is commonly referred to as the ''S''-polynomial, where ''S'' refers to ''subtraction'' (Buchberger) or ''[[Syzygy (mathematics)|Syzygy]]'' (others). The pair of polynomials to which it is associated is commonly referred to as [[critical pair (
There are numerous ways to improve this algorithm beyond what has been stated above. For example, one could reduce all the new elements of ''F'' relative to each other before adding them. It also should be noted that if the leading terms of ''f<sub>i</sub>'' and ''f<sub>j</sub>'' share no variables in common, then ''S<sub>ij</sub>'' will ''always'' reduce to 0 (if we use only f<sub>i</sub> and f<sub>j</sub> for reduction), so we needn't calculate it at all.
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