§ Bucchberger algorithm
- multidegree: term of maximum degree, where maximum is defined via lex ordering.
- Alternatively, multidegree is the degree of the leading term.
multideg(f) = a and
multideg(g) = b, define
c[i] = max(a[i], b[i]). Then is the LCM of the leading monomial of and the leading monomial of .
- The S-polynomial of and is the combination
- The S-polynomial is designed to create cancellations of leading terms.
§ Bucchberger's criterion
- Let be an ideal. Then a basis is a Groebner basis iff for all pairs , .
- Recall that a basis is an Grober basis iff . That is, the ideal of leading terms of is generated by the leading terms of the generators.
- for a basis , we should consider . If , then make .
- Repeat till we find that