Cox-Little-O’Shea 2.5 - The Hilbert Basis Theorem and Gröbner Bases
Definition 1 (ideal of leading terms)
[definition] Let $I \subseteq k[x_1, \ldots, x_n]$ be a nonzero ideal and fix a monomial ordering on $k[x_1, \ldots, x_n]$. Then
(i) We denote by $\operatorname{LT}(I)$ as
\[\operatorname{LT}(I) = \{cx^\alpha \mid \text{there exists } f\in I \setminus 0 \text{ with } \operatorname{LT}(f) = cx^\alpha\}.\](ii) We denote by $\langle \operatorname{LT}(I)\rangle$ the ideal generated by the terms of $\operatorname{LT}(I)$. [/definition]
Proposition 3
[proposition] Let $I \subseteq k[x_1, \ldots, x_n]$ be a nonzero ideal.
(i) $\langle \operatorname{LT}(I)\rangle$ is a monomial ideal.
(ii) There are $g_1, \ldots, g_t \in I$ such that
\[\langle \operatorname{LT}(I)\rangle = \langle \operatorname{LT}(g_1), \ldots, \operatorname{LT}(g_t)\rangle.\][/proposition]
Definition 5 (Grobner basis)
[definition] Fix a monomial order on the polynomial ring $k[x_1, \ldots, x_n]$. A finite subset $G = \{g_1, \ldots, g_t\}$ of a nonzero ideal $I \subseteq k[x_1, \ldots, x_n]$ is a Grobner basis if
\[\langle \operatorname{LT}(I)\rangle = \langle \operatorname{LT}(g_1), \ldots, \operatorname{LT}(g_t)\rangle.\][/definition]
Corollary 6
[corollary] Fix a monomial order. Then every ideal $I \subseteq k[x_1, \ldots, x_n]$ has a Grobner basis. Furthermore, any Grobner basis for an ideal $I$ is a basis of $I$ (hence the name). [/corollary]