Monomials

First, what is a ``term'' of a polynomial of several variables? Let $x_1,...,x_n$ be variables and let $R$ be a ring. A monomial (or ``term'') is a polynomial of the form

$$x_1^{i_1}...x_n^{i_n},$$

where the $i_j$'s are non-negative integers. This expression is often abreviated using ``multi-index notation'' as $\underline{x}^{\underline{i}}$, where $\underline{i}=(i_1,...,i_n)$ and $\underline{x}=(x_1,...,x_n)$ and is a vector of variables. Note that the map

$$\begin{array}{cccc} \phi:&\mathbb{N}^n &\rightarrow &\{{\rm ... ... & (i_1,...,i_n)&\longmapsto &x_1^{i_1}...x_n^{i_n} \end{array}$$

defines a bijection from the set of $n$-tuples of non-negative integers to the set of monomials. In other words, there is a 1-1 correspondence between monomials and their vectors of exponents.

Though there may be some ambiguity as to how to divide one arbitrary polynomial by another, for the reasons discuessed above, when we reastrict our considerations to monomials, the ambiguities disappear. If $R$ is a ring and

$$f(x_1,...,x_n)=ax_1^{a_1}...x_n^{a_n}\in R[x_1,...,x_n], \ \ \ g(x_1,...,x_n)=bx_1^{b_1}...x_n^{b_n}\in R[x_1,...,x_n],$$

are monomials ($a_i\geq 0$, $b_j\geq 0$ are non-negative intgers) then $f$ divides $g$, written $f\vert g$ if and only if $a\vert b$ in $R$ and $a_i\leq b_i$ for all $1\leq i\leq n$.