Monomial

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.
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 also that no obvious other notion is available to designate these values (the term power product is in use, in particular when "monomial" is used with the first meaning, but it does not make the absence of constants clear either), 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"."
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_8 in formula_9 variables is the number of multicombinations of formula_8 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_12. This expression can also be given in the form of a binomial coefficient, as a polynomial expression in formula_8, or using a rising factorial power of formula_14:
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_8 of degree formula_17 with leading coefficient formula_18.
For example, the number of monomials in three variables (formula_19) of degree "d" is formula_20; 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_8 in formula_9 variables is the coefficient of degree formula_8 of the formal power series expansion of
The number of monomials of degree at most in variables is formula_25. This follows from the one-to-one correspondence between the monomials of degree formula_8 in formula_27 variables and the monomials of degree at most formula_8 in formula_9 variables, which consists in substituting by 1 the extra variable.
Notation.
Notation for monomials is constantly required in fields like partial differential equations. If the variables being used form an indexed family like formula_30, formula_31, formula_32, ..., then "multi-index notation" is helpful: if we write 
we can define 
for compactness.
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_35. The degree of formula_36 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_37 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".