Polynomials

Suppose there are $n$ variables, $x_1, x_2, \ldots, x_n$. A monomial over a field ${\mathbb{F}}$ is a product of the form

$$x_1^{d_1} \cdot x_2^{d_2} \cdots \cdot x_n^{d_n} ,$$ (4.47)

in which all of the exponents $d_1$, $d_2$, $\ldots$, $d_n$ are positive integers. The total degree of the monomial is $d_1 + \cdots + d_n$.

A polynomial $f$ in variables $x_1,\ldots, x_n$ with coefficients in ${\mathbb{F}}$ is a finite linear combination of monomials that have coefficients in ${\mathbb{F}}$. A polynomial can be expressed as

$$\sum_{i = 1}^m c_i m_i ,$$ (4.48)

in which $m_i$ is a monomial as shown in (4.47), and $c_i \in {\mathbb{F}}$ is a coefficient. If $c_ i \not = 0$, then each $c_i m_i$ is called a term. Note that the exponents $d_i$ may be different for every term of $f$. The total degree of $f$ is the maximum total degree among the monomials of the terms of $f$. The set of all polynomials in $x_1,\ldots, x_n$ with coefficients in ${\mathbb{F}}$ is denoted by ${\mathbb{F}}[x_1,\ldots, x_n]$.

Example 4..17 (Polynomials)   The definitions correspond exactly to our intuitive notion of a polynomial. For example, suppose ${\mathbb{F}}= {\mathbb{Q}}$. An example of a polynomial in ${\mathbb{Q}}[x_1,x_2,x_3]$ is

$$x_1^4 - \begin{matrix}\frac{1}{2} \end{matrix} x_1 x_2 x_3^3 + x_1^2 x_2^2 + 4 .$$ (4.49)

Note that $1$ is a valid monomial; hence, any element of ${\mathbb{F}}$ may appear alone as a term, such as the $4 \in {\mathbb{Q}}$ in the polynomial above. The total degree of (4.49) is $5$ due to the second term. An equivalent polynomial may be written using nicer variables. Using $x$, $y$, and $z$ as variables yields

$$x^4 - \begin{matrix}\frac{1}{2} \end{matrix} x y z^3 + x^2 y^2 + 4 ,$$ (4.50)

which belongs to ${\mathbb{Q}}[x,y,z]$. $\blacksquare$

The set ${\mathbb{F}}[x_1,\ldots, x_n]$ of polynomials is actually a group with respect to addition; however, it is not a field. Even though polynomials can be multiplied, some polynomials do not have a multiplicative inverse. Therefore, the set ${\mathbb{F}}[x_1,\ldots, x_n]$ is often referred to as a commutative ring of polynomials. A commutative ring is a set with two operations for which every axiom for fields is satisfied except the last one, which would require a multiplicative inverse.