Monomial

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In mathematics, a monomial (or mononomial) is a particular kind of polynomial, having just one term. Given a natural number n and a variable x, the power function defined by the rule f(x)=xⁿ 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^ay^bz^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., 7x^ay^bz^c).

1 As bases
2 Notation
3 Geometry
4 Other uses
5 See also

[edit] As bases

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-Szász theorem).

[edit] Notation

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^a x₂^b x₃^c

and save a great deal of space.

[edit] Geometry

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.