A monomial is a product of positive integer powers of a fixed set of variables (possibly) together with a coefficient, e.g., 
, 
,
or 
. A monomial can also be thought
of as a nonzero summand of a polynomial (Becker and
Weispfenning 1993, p. 191; Cox et al. 1996). A monomial with the coefficient
excluded is usually called a term.
Unfortunately, in some older works, the definitions of monomial and term are sometimes reversed. Care is therefore needed in attempting to distinguish these conflicting usages.
The Wolfram Language command MonomialList[poly,
, 
,
...
] gives the list of monomials with respect
to the variables 
in the specified polynomial.
The monomials 
and 
are orthogonal on the unit circle
in the complex plane (Dumitriu et
al. 2004) since
 |
(1)
|
The monomial functions 
are defined as
 |
(2)
|
where 
is the set of permutations giving
distinct terms in the sum and 
is considered to be infinite (Dumitriu et al. 2004).
For example.
 |
(3)
|
Care is needed when consulting the literature, since the distinction between terms and monomials is not always observed. For example, Dummit and Foote (1998, p. 234) define a monomial as a polynomial with only one nonzero term, without defining what is meant by "term."
