§ Bucchberger algorithm

• multidegree: term of maximum degree, where maximum is defined via lex ordering.
• Alternatively, multidegree is the degree of the leading term.
• If multideg(f) = a and multideg(g) = b, define c[i] = max(a[i], b[i]). Then $\vec x^c$ is the LCM of the leading monomial of $f$ and the leading monomial of $g$.
• The S-polynomial of $f$ and $g$ is the combination $\vec (x^c/LT(f)) f - (\vec x^c/LT(g)) g$
• The S-polynomial is designed to create cancellations of leading terms.

§ Bucchberger's criterion

• Let $I$ be an ideal. Then a basis $\langle g_1, \dots, g_N \rangle$ is a Groebner basis iff for all pairs $i \neq j$, $S(g_i, g_j) = 0$.
• Recall that a basis is an Grober basis iff $LT(I) = \langle LT(g_1), \dots, LT(g_N) \rangle$. That is, the ideal of leading terms of $I$is generated by the leading terms of the generators.

• for a basis $F$, we should consider $r(i, j) \equiv rem_F(S(f_i, f_j))$. If $r(i, j) \neq 0$, then make $F' \equiv F \cup \{ S(f_i, f_j \}$.
• Repeat till we find that