Gröbner basis

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

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

2. $1\le a$ for all $a\in S$.

In practice, for any $a=x_1^{a_1}{x}_2^{a_2}\mathrm{\cdots }{x}_n^{a_n}\in F[x_1,\mathrm{\dots },x_n]$, we associate to $a$ the string $(a_1,a_2,\mathrm{\dots },a_n)$ 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 $\mathrm{supp}(a)$, to be the set of terms $x_i$ with $a_i\ne 0$. Then define $M(a)=\max (\mathrm{supp}(a))$.

A partial order on $F\mathbf{}\mathrm{[}{x}_{\mathrm{1}}\mathrm{,}\mathrm{\dots }\mathrm{,}{x}_n\mathrm{]}$:
We can extend our monomial ordering to a partial ordering on $F[x_1,\mathrm{\dots },x_n]$ as follows:Let $a,b\in F[x_1,\mathrm{\dots },x_n]$. If $\mathrm{supp}(a)\ne \mathrm{supp}(b)$, we say that if .

It can be shown that:

1. The relation defined above is indeed a partial order on $F[x_1,\mathrm{\dots },x_n]$

2. Every descending chain $p_1(x_1,\mathrm{\dots },x_n)>p_2(x_1,\mathrm{\dots },x_n)>\mathrm{\dots }$ with $p_i\in [x_1,\mathrm{\dots },x_n]$ is finite.

A division algorithm for $F\mathbf{}\mathrm{[}{x}_{\mathrm{1}}\mathrm{,}\mathrm{\dots }\mathrm{,}{x}_n\mathrm{]}$:
We can then formulate a division algorithm for $F[x_1,\mathrm{\dots },x_n]$:
Let $(f_1,\mathrm{\dots },f_s)$ be an ordered $s$-tuple of polynomials, with $f_i\in F[x_1,\mathrm{\dots },x_n]$. Then for each $f\in F[x_1,\mathrm{\dots },x_n]$, there exist $a_1,\mathrm{\dots },a_s,r\in F[x_1,\mathrm{\dots },x_n]$ with $r$ unique, such that

1. $f=a_1{f}_1+\mathrm{\cdots }+a_s{f}_s+r$

2. For each $i=1,\mathrm{\dots },s$, $M(a_i)$ does not divide any monomial in $\mathrm{supp}(r)$.

Furthermore, if $a_i{f}_i\ne 0$ for some $i$, then $M(a_i{f}_i)\le M(f)$.

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

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