Polynomial with only one term
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
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.
In mathematical analysis, it is common to consider polynomials written in terms of a shifted variable formula_19 for some constant formula_20 rather than a variable formula_4 alone, as in the study of Taylor series. By a slight abuse of notation, monomials of shifted variables, for instance formula_22 may be called monomials in the sense of shifted monomials or centered monomials, where formula_20 is the center or formula_24 is the shift.
Since the word "monomial", as well as the word "polynomial", comes from the late Latin word "binomium" (binomial), by changing the prefix "bi-" (two in Latin), a monomial should theoretically be called a "mononomial". "Monomial" is a syncope by haplology 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. 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 that no obvious other notion is available to designate these values, though primitive monomial is in use and does make the absence of constants clear. 
"The remainder of this article assumes the first meaning of "monomial"."
Monomial basis.
The most obvious fact about monomials (first meaning) is that any polynomial is a linear combination of them, so they form a basis of the vector space of all polynomials, called the "monomial basis" - a fact of constant implicit use in mathematics.
Number.
The number of monomials of degree formula_25 in formula_9 variables is the number of multicombinations of formula_25 elements chosen among the formula_9 variables (a variable can be chosen more than once, but order does not matter), which is given by the multiset coefficient formula_29. This expression can also be given in the form of a binomial coefficient, as a polynomial expression in formula_25, or using a rising factorial power of formula_31:
formula_32
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 formula_25 of degree formula_34 with leading coefficient formula_35.
For example, the number of monomials in three variables (formula_36) of degree "d" is formula_37; 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 formula_25 in formula_9 variables is the coefficient of degree formula_25 of the formal power series expansion of
formula_41
The number of monomials of degree at most "d" in "n" variables is formula_42. This follows from the one-to-one correspondence between the monomials of degree formula_25 in formula_44 variables and the monomials of degree at most formula_25 in formula_9 variables, which consists in substituting by 1 the extra variable.
Multi-index notation.
The "multi-index notation" is often useful for having a compact notation, specially when there are more than two or three variables. If the variables being used form an indexed family like formula_47 one can set 
formula_48
and
formula_49
Then the monomial 
formula_50
can be compactly written as
formula_51
With this notation, the product of two monomials is simply expressed by using the addition of exponent vectors:
formula_52
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 formula_53. The degree of formula_54 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 formula_55 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 (equivalently by a multiplicative group of diagonal matrices). This area is studied under the name of "torus embeddings".
References.
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