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). Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... In elementary mathematics, a term is either a single number or variable, or the product of several numbers and/or variables. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In mathematics, exponentiation is a process generalized from repeated multiplication, in much the same way that multiplication is a process generalized from repeated addition. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...

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). In mathematics, a coefficient is a multiplicative factor of a certain object such as a variable (for example, the coefficients of a polynomial), a basis vector, a basis function and so on. ...

As bases

The most obvious fact about momomials 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). In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... In mathematical analysis, the uniform norm assigns to real- or complex-valued functions f the nonnegative number This norm is also called the supremum norm or the Chebyshev norm. ... In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on an interval [a,b] can be uniformly approximated as closely as desired by a polynomial function. ... In mathematics, the Müntz–Szász theorem is a basic result of approximation theory, proved by Herman Müntz in 1914 and Otto Szász (1884-1952) in 1916. ...

Notation

Notation for monomials is constantly required in fields like partial differential equations. Multi-index notation is helpful: if we write In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... The notion of multi-indices simplifies formulae used in the multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an array of indices. ...

α = (a, b, c)

we can define

x^α = x₁^a x₂^b x₃^c

and save a great deal of space.

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. Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety. ... In mathematics, a symmetry group describes all symmetries of objects. ... In mathematics, an algebraic torus over a field K is an algebraic group which is isomorphic over the algebraic closure of K to (GL1)r for some integer r, the rank of the torus. ... In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. ... In mathematics and theoretical physics, toric geometry is a set of methods in algebraic geometry in which complex manifolds are visualized as fiber bundles with multi-dimensional tori as fibers. ...

See also

• binomial
• homogeneous polynomial
• homogeneous function
• multilinear form