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 t^{n} 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 matrices). This area is studied under the name of *torus embeddings*.

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