Definition of monomial orderings and support:
Let $F$ be a field, and let $S$ be the set of monomials in $F[x_1,\ldots,x_n]$ , the polynomial ring in $n$ indeterminates. A monomial ordering is a total ordering $\leq$ on $S$ which satisfies

1. $a \leq b$ implies that $ac \leq bc$ for all $a,b,c \in S$ .
2. $1 \leq a$ for all $a \in S$ .

In practice, for any $a=x_1^{a_1}x_2^{a_2}\cdots x_n^{a_n} \in F[x_1,\ldots,x_n]$ , we associate to $a$ the string $(a_1,a_2,\ldots,a_n)$ and compare monomials by looking at orderings on these $n$ -tuples.

Example. An extremely prevalent example of a monomial ordering is given by the standard lexicographical ordering of strings. Other examples include graded lexicographic ordering and graded reverse lexicographic ordering.

Henceforth, assume that we have fixed a monomial ordering. Define the support of $a$ , denoted $\supp(a)$ , to be the set of terms $x_i$ with $a_i\neq 0$ . Then define $M(a)=\max(\supp(a))$ .

A partial order on $F[x_1,\ldots,x_n]$ :
We can extend our monomial ordering to a partial ordering on $F[x_1,\ldots,x_n]$ as follows: Let $a,b \in F[x_1,\ldots,x_n]$ . If $\supp(a) \neq \supp(b)$ , we say that $a < b$ if $\max(\supp(a)-\supp(b)) < \max(\supp(b)-\supp(a))$ .

It can be shown that:

1. The relation defined above is indeed a partial order on $F[x_1,\ldots,x_n]$
2. Every descending chain $p_1(x_1,\ldots,x_n) > p_2(x_1,\ldots,x_n) > \ldots$ with $p_i \in [x_1,\ldots,x_n]$ is finite.

A division algorithm for $F[x_1,\ldots,x_n]$ :
We can then formulate a division algorithm for $F[x_1,\ldots,x_n]$ :
Let $(f_1,\ldots,f_s)$ be an ordered $s$ -tuple of polynomials, with $f_i \in F[x_1,\ldots,x_n]$ . Then for each $f \in F[x_1,\ldots,x_n]$ , there exist $a_1,\ldots,a_s, r \in F[x_1,\ldots,x_n]$ with $r$ unique, such that

1. $f=a_1f_1+\cdots+a_sf_s+r$
2. For each $i=1,\ldots,s$ , $M(a_i)$ does not divide any monomial in $\supp(r)$ .

Definition of Gröbner basis:
Let $I$ be a nonzero ideal of $F[x_1,\ldots,x_n]$ . A finite set $T \subset I$ of polynomials is a Gröbner basis for $I$ if for all $b \in I$ with $b \neq 0$ there exists $p \in T$ such that $M(p) \mid M(b)$ .

Existence of Gröbner bases:
Every ideal $I \subset k[x_1,\ldots,x_n]$ other than the zero ideal has a Gröbner basis. Additionally, any Gröbner basis for $I$ is also a basis of $I$ .