In mathematics, a monomial is a particular kind of polynomial, having just one term (mathematics). Given a natural number n and a variable x , the power function defined by the rule f(x)=x n is therefore a monomial. Given several unknown variables (say, x , y , z ) and corresponding natural number exponents (say, a, b, c), the product of the resulting univariate monomials is also a monomial (e.g., the function determined by the rule f(x)=x a y b z c).

If Coefficients are allowed (this may not be consistent), then a constant multiple of a monomial is also counted as a monomial (e.g., 7 x a y b z c).

The most obvious fact about monomials is that any polynomial is a linear combination of them, so they can serve as basis vectors in a vector space of polynomials - a fact of constant implicit use in mathematics. An interesting fact from functional analysis is that the full set of monomials tn is not required to span a linear subspace of C[0,1] that is dense for the uniform norm (sharpening the Stone-Weierstrass theorem). It is enough that the sum of the reciprocals n-1 diverge (the Müntz-Szasz theorem).

Notation for monomials is constantly required in fields like partial differential equations. Multi-index notation is helpful: if we write

:α = ( a , b , c )

we can define

: x α = x 1 a x 2 b x 3 c

and save a great deal of space.

In algebraic geometry the varieties defined by monomial equations x α = 0 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 matrix). This area is studied under the name of torus embeddings .

In group representation theory, a monomial representation is a particular kind of induced representation.

In propositional logic, a Monomial is a logical conjunction of literals. (See also Clause, Minterm.)

=See also=