In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two
different definitions of a monomial may be encountered:
For the first definition, a monomial is a product of powers of variables with nonnegative
integer exponents, or, in other words, a product of variables, possibly with repetitions. The
constant 1 is a monomial, being equal to the empty product and x0 for any variable x. If
only a single variable x is considered, this means that a monomial is either 1 or a power
xn of x, with n a positive integer. If several variables are considered, say, , , , then
each can be given an exponent, so that any monomial is of the form with non-negative
integers. For the second definition, a monomial is a
monomial in the first sense multiplied by a nonzero constant, called the coefficient
of the monomial. A monomial in the first sense is also a monomial in the second sense, because
the multiplication by 1 is allowed. For example, in this interpretation and are monomials.
In the context of Laurent polynomials and Laurent series, the exponents of a monomial
may be negative, and in the context of Puiseux series, the exponents may be rational numbers.
Since the word "polynomial" comes from "poly-" plus the Greek word "νομός", a monomial
should theoretically be called a "mononomial". "Monomial" is a syncope of "mononomial".
Comparison of the two definitions With either definition, the set of monomials
is a subset of all polynomials that is closed under multiplication.
Both uses of this notion can be found, and in many cases the distinction is simply ignored,
see for instance examples for the first and second meaning, and an unclear definition.
In informal discussions the distinction is seldom important, and tendency is towards
the broader second meaning. When studying the structure of polynomials however, one
often definitely needs a notion with the first meaning. This is for instance the case when
considering a monomial basis of a polynomial ring, or a monomial ordering of that basis.
An argument in favor of the first meaning is also that no obvious other notion is available
to designate these values, while the notion term of a polynomial unambiguously coincides
with the second meaning of monomial. The remainder of this article assumes the
first meaning of "monomial". As bases
The most obvious fact about monomials is that any polynomial is a linear combination of
them, so they form a basis of the vector space of all polynomials - a fact of constant implicit
use in mathematics. Number
The number of monomials of degree d in n variables is the number of multicombinations of d elements
chosen among the n variables, which is given by the multiset coefficient . This expression
can also be given in the form of a binomial coefficient, as a polynomial expression in
d, or using a rising factorial power of d + 1:
The latter forms are particularly useful when one fixes the number of variables and lets
the degree vary. From these expressions one sees that for fixed n, the number of monomials
of degree d is a polynomial expression in d of degree with leading coefficient .
For example, the number of monomials in three variables of degree d is ; these numbers form
the sequence 1, 3, 6, 10, 15, ... of triangular numbers.
The Hilbert series is a compact way to express the number of monomials of a given degree:
the number of monomials of degree d in n variables is the coefficient of degree d of the formal
power series expansion of
Notation Notation for monomials is constantly required
in fields like partial differential equations. If the variables being used form an indexed
family like , , , ..., then multi-index notation is helpful: if we write
we can define
and save a great deal of space. Degree
The degree of a monomial is defined as the sum of all the exponents of the variables,
including the implicit exponents of 1 for the variables which appear without exponent;
e.g., in the example of the previous section, the degree is . The degree of is 1+1+2=4.
The degree of a nonzero constant is 0. For example, the degree of -7 is 0.
The degree of a monomial is sometimes called order, mainly in the context of series. It
is also called total degree when it is needed to distinguish it from the degree in one of
the variables. Monomial degree is fundamental to the theory
of univariate and multivariate polynomials. Explicitly, it is used to define the degree
of a polynomial and the notion of homogeneous polynomial, as well as for graded monomial
orderings used in formulating and computing Gröbner bases. Implicitly, it is used in
grouping the terms of a Taylor series in several variables.
Geometry In algebraic geometry the varieties defined
by monomial equations for some set of α have special properties of homogeneity. This can
be phrased in the language of algebraic groups, in terms of the existence of a group action
of an algebraic torus. This area is studied under the name of torus embeddings.
See also Monomial representation
Monomial matrix Homogeneous polynomial
Homogeneous function Multilinear form
Log-log plot Power law