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Monomial

[moh-noh-mee-uhl, muh-]
In mathematics, the word monomial means two different things in the context of polynomials:

• The first meaning is a product of powers of variables, or formally any value obtained from 1 by finitely many multiplications by a variable. If only a single variable $x$ is considered this means that any monomial is either 1 or a power $x^n$ of $x$, with $n$ a positive integer. If several variables are considered, say, $x$, $y$, $z$, then each can be given an exponent, so that any monomial is of the form $x^a y^b z^c$ with $a,b,c$ nonnegative integers (taking note that any exponent 0 makes the corresponding factor equal to 1).
• The second meaning of monomial includes monomials in the first sense, but also allows multiplication by any constant, so that $-7x^5$ and $\left(3-4i\right)x^4yz^\left\{13\right\}$ are also considered to be monomials (the second example assuming polynomials in $x$, $y$, $z$ over the complex numbers are considered).

With either definition, the set of monomials is a subset of all polynomials that is closed under multiplication.

Both uses of this notion can be found, and in many cases the distinction is simply ignored, see for instance examples for the first and second meaning, and an unclear definition In informal discussions the distinction is seldom important, and tendency is towards the broader second meaning. When studying the structure of polynomials however, one often definitely needs a notion with the first meaning. This is for instance the case when considering a monomial basis of a polynomial ring, or a monomial ordering of that basis. An argument in favor of the first meaning is also that no obvious other notion is available to designate these values (the term power product is in use, but it does not make the absence of constants clear either), while the notion $term$ of a polynomial unambiguously coindices with the second meaning of monomial. For an isolated polynomial consisting of a single term, one could if necessary use the uncontracted form mononomial, analogous to binomial and trinomial. The remainder of this article assumes the first meaning of "monomial".

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).

Notation

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

$alpha = \left(a, b, c\right)$

we can define

$x^\left\{alpha\right\} = x_1^a, x_2^b, x_3^c$

and save a great deal of space.

Geometry

In algebraic geometry the varieties defined by monomial equations $x^\left\{alpha\right\} = 0$ 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.

Linear Algebra

In linear algebra a monomial matrix is usually defined as a square matrix having one and only one non-zero element per row and per column. In other words, it can be obtained through the multiplication of a permutation matrix and a (regular) diagonal matrix. More generally, a rectangular monomial matrix is a matrix with one and only one non-zero element per row and at most one per column; the rectangular monomial matrices are composed from distinct rows of some square monomial matrices. The two matrices A and B below are 3-by-3 and 2-by3 monomial matrices, respectively.

$A=begin\left\{bmatrix\right\} 1 & 0 & 0 0 & 0 & -0.5 0 & 35 & 0 end\left\{bmatrix\right\}=begin\left\{bmatrix\right\} 1 & 0 & 0 0 & 0 & 1 0 & 1 & 0 end\left\{bmatrix\right\}.begin\left\{bmatrix\right\} 1 & 0 & 0 0 & 35 & 0 0 & 0 & -0.5 end\left\{bmatrix\right\},$

B=begin{bmatrix} 1 & 0 & 0 0 & 0 & -0.5 end{bmatrix}