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

in which all of the exponents , , , are positive integers. The

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

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