Monomial

In mathematics, in the context of polynomials, the word monomial can have one of two different meanings:

• The first is a product of powers of variables, or formally any value obtained by finitely many multiplications of a variable. If only a single variable x is considered, this means that any monomial is either 1 or a power xⁿ 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^ay^bz^c with a,b,c non-negative 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 − 7x⁵ and (3 − 4i)x⁴yz¹³ are also considered to be monomials (the second example assuming polynomials in x, y, z over the complex numbers are considered).

Comparison of the two definitions

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^[1] and second^[2] 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 coincides 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 (first meaning) is that any polynomial is a linear combination of them, so they form a basis of the vector space of all polynomials - a fact of constant implicit use in mathematics.

Number

The number of monomials of degree d in n variables is the number of multicombinations of d elements chosen among the n variables (a variable can be chosen more than once, but order does not matter), which is given by the multiset coefficient $\textstyle{\left(\!\!{n\choose d}\!\!\right)}$. This expression can also be given in the form of a binomial coefficient, as a polynomial expression in d, or using a rising factorial power of d + 1:

$\left(\!\!{n\choose d}\!\!\right) = \binom{n+d-1}{d} = \binom{d+(n-1)}{n-1} = \frac{(d+1)\times(d+2)\times\cdots\times(d+n-1)}{1\times2\times\cdots\times(n-1)} = \frac{1}{(n-1)!}(d+1)^{\overline{n-1}}.$

The latter forms are particularly useful when one fixes the number of variables and lets the degree vary. From these expressions one sees that for fixed n, the number of monomials of degree d is a polynomial expression in d of degree n − 1 with leading coefficient $\tfrac1{(n-1)!}$.

For example, the number of monomials in three variables (n = 3) of degree d is $\textstyle{\frac{1}{2}}(d+1)^{\overline2} = \textstyle{\frac{1}{2}}(d+1)(d+2)$; these numbers form the sequence 1, 3, 6, 10, 15, ... of triangular numbers.

Notation

Notation for monomials is constantly required in fields like partial differential equations. If the variables being used form an indexed family like x₁, x₂, x₃, ..., then multi-index notation is helpful: if we write

α = (a,b,c)

we can define

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

and save a great deal of space.

Geometry

In algebraic geometry the varieties defined by monomial equations x^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.

See also

• Monomial representation
• Monomial matrix
• Homogeneous polynomial
• Homogeneous function
• Multilinear form

Notes

1. ^ Cox, David; John Little, Donal O'Shea (1998). Using Algebraic Geometry. Springer Verlag. pp. 1. ISBN 0-387-98487-9. 
2. ^ Hazewinkel, Michiel, ed. (2001), "Monomial", Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/M/m064760.htm