Monomials

Definition: Monomial

A monomial over a commutative ring $R$ in the variables $x_1,\cdots ,x_n$ is an expression of the form

$$a{x}_n^{p_n}{x}_{n-1}^{p_{n-1}}\cdots x_2^{p_2}{x}_1^{p_1},$$

where $a\in R,a\ne 0_R$ and $p_1,\cdots ,p_n$ are non-negative integers.

NOTATION

If $p_i=0$, then we do not write $x_i^0$ or $1$, we just do not write anything about $x_i$ at all.

If $p_i=1$, then we write simply $x_i$ instead of $x_i^1$.

We usually use capital Latin letters such as $M$ and $N$ to denote monomials. If we want to be explicit about the variables, we write $M(x_1,\cdots ,x_n)$.

Definition: Coefficient of a Monomial

We call $a$ the coefficient of the monomial.

Degree

Definition: Degree of a Monomial

The degree of a monomial $M=a{x}_n^{p_n}{x}_{n-1}^{p_{n-1}}\cdots x_2^{p_2}{x}_1^{p_1}$ is $\max \{p_1,\dots ,p_n\}$.

NOTATION

$$\mathrm{deg}(M)$$

EXAMPLE

$$\mathrm{deg}(4x^4{y}^3{z}^5)=5$$ $$\mathrm{deg}(9y^{2}z)=2$$ $$\mathrm{deg}(xy)=1$$

Equality

Definition: Equality of Monomials

Two monomials $M=m{x}_n^{p_n}{x}_{n-1}^{p_{n-1}}\cdots x_2^{p_2}{x}_1^{p_1}$ and $M^{\prime }=m^{\prime }{x}_n^{p_n^{\prime }}{x}_{n-1}^{p_{n-1}^{\prime }}\cdots x_2^{p_2^{\prime }}{x}_1^{p_1^{\prime }}$ are equal if $m=m^{\prime }$ and $p_k=p_k^{\prime }$ for all $k\in \{1,\cdots ,n\}$.

NOTE

The order of the variables $x_1,\cdots ,x_n$ is irrelevant.

NOTATION

$$M=M^{\prime }$$

EXAMPLE

$$3x^2{y}^3=3y^3{x}^2$$ $$4x^4{y}^{2}z=4z{x}^4{y}^2$$

Operations

Definition: Monomial Addition

The sum of two monomials $M=m{x}_n^{p_n}{x}_{n-1}^{p_{n-1}}\cdots x_2^{p_2}{x}_1^{p_1}$ and $M^{\prime }=m^{\prime }{x}_n^{p_n}{x}_{n-1}^{p_{n-1}}\cdots x_2^{p_2}{x}_1^{p_1}$ is the monomial

$$M+M^{\prime }\stackrel{\text{def}}{=}(m+m^{\prime })x_n^{p_n}{x}_{n-1}^{p_{n-1}}\cdots x_2^{p_2}{x}_1^{p_1}$$

EXAMPLE

$$2x^{3}y+3x^{3}y=5x^{3}y$$ $$2.5x_1^4{x}_2^2{y}^2+1.3x_1^4{x}_2^2{y}^2=3.8x_1^4{x}_2^2{y}^2$$

Definition: Monomial Multiplication

Let $R$ be a commutative ring and let $M=m{x}_n^{p_n}{x}_{n-1}^{p_{n-1}}\cdots x_2^{p_2}{x}_1^{p_1}$ and $M^{\prime }=m^{\prime }{x}_n^{p_n^{\prime }}{x}_{n-1}^{p_{n-1}^{\prime }}\cdots x_2^{p_2^{\prime }}{x}_1^{p_1^{\prime }}$ be two monomials over $R$.

The product of a $M$ with $M^{\prime }$ is defined as the monomial

$$\begin{array}{r}M{M}^{\prime }\stackrel{\text{def}}{=}(m{m}^{\prime })x_n^{p_n+p_n^{\prime }}{x}_{n-1}^{p_{n-1}+p_{n-1}^{\prime }}\cdots x_2^{p_2+p_2^{\prime }}{x}_1^{p_1+p_1^{\prime }}\end{array}$$

INTUITION

Monomial multiplication is defined in this in order to follow the commutativity and associativity laws for the underlying ring.

EXAMPLE

$$(3x^{4}y)\cdot (2x^{4}y)=(2\cdot 3)x^{4+4}{y}^{1+1}=6x^8{y}^2$$ $$(11x^3{y}^{2}z)\cdot (2y{z}^2)=(11x^3{y}^{2}z)\cdot (2x^{0}y{z}^2)=(11\cdot 2)(x^{3+0}{y}^{2+1}{z}^{1+2})=22x{y}^3{z}^3$$ $$(x^{3}y)(y{z}^3)=(x^{3}y{z}^0)(x^{0}y{z}^3)=(1\cdot 1)(x^{3+0}{y}^{1+1}{z}^{0+3})=x^3{y}^2{z}^3$$