A monomial is a product of nonnegative integer powers of a fixed set of variables,
possibly together with a nonzero coefficient,
e.g., 
, 
, or 
. A monomial can also be thought of as a nonzero
summand of a polynomial (Becker and Weispfenning 1993,
p. 191). A monomial with the coefficient excluded is usually called a term.
Another common convention reserves "monomial" for the product with its coefficient omitted and uses "term" for a summand including its coefficient (Cox et al. 1997, p. 2). This convention is used in the definitions of leading monomial, leading term, and polynomial reduction. It is necessary to specify the convention when comparing sources.
The Wolfram Language command MonomialList[poly,
, 
,
...
] gives the list of monomials with respect
to the variables 
in the specified polynomial.
The monomials 
form an orthogonal family on the unit
circle in the complex plane with respect to
angular measure (Dumitriu et al. 2007). For nonnegative
integers 
and 
, writing 
gives
 |
(1)
|
where 
is the Kronecker
delta.
The monomial symmetric polynomials 
, also called monomial functions,
sum all distinct monomials obtained by permuting an exponent list. For a partition 
with at most 
parts, pad it with zeros to length 
and write
 |
(2)
|
where 
is the set
of distinct permutations of that list (Dumitriu et
al. 2007). For example, in three variables,
 |
(3)
|
Care is needed when consulting the literature, since the distinction between terms and monomials is not always observed. For example, Dummit and Foote (2004, p. 234) define a monomial as a polynomial with only one nonzero term, without defining what is meant by "term."
