Monomial

In mathematics, a monomial is a particular kind of polynomial, having just one term. The monomials in one unknown X are therefore the powers Xⁿ for n = 0, 1, 2, ... . Given several unknown variables X, Y, Z a monomial would take the form X^aY^bZ^c for some natural numbers a, b and c. Whether or not coefficients are allowed may not be consistent: 7X^aY^bZ^c might sometimes be counted as a monomial.

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 = X₁^aX₂^bX₃^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.