#### Polynomials

Suppose there are variables, . A monomial over a field is a product of the form

 (4.47)

in which all of the exponents , , , are positive integers. The total degree of the monomial is .

A polynomial in variables with coefficients in is a finite linear combination of monomials that have coefficients in . A polynomial can be expressed as

 (4.48)

in which is a monomial as shown in (4.47), and is a coefficient. If , then each is called a term. Note that the exponents may be different for every term of . The total degree of is the maximum total degree among the monomials of the terms of . The set of all polynomials in with coefficients in is denoted by .

Example 4..17 (Polynomials)   The definitions correspond exactly to our intuitive notion of a polynomial. For example, suppose . An example of a polynomial in is

 (4.49)

Note that is a valid monomial; hence, any element of may appear alone as a term, such as the in the polynomial above. The total degree of (4.49) is due to the second term. An equivalent polynomial may be written using nicer variables. Using , , and as variables yields

 (4.50)

which belongs to .

The set 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 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.

Steven M LaValle 2020-08-14