# Monomial

In mathematics, a monomial is, roughly speaking, a polynomial which has only one term.

Two definitions of a monomial may be encountered:

In the context of Laurent polynomials and Laurent series, the exponents of a monomial may be negative, and in the context of Puiseux series, the exponents may be rational numbers.

Since the word "monomial", as well as the word "polynomial", comes from the late Latin word "binomium" (binomial), by changing the prefix "bi" (two in Latin), a monomial should theoretically be called a "mononomial".

"Monomial" is a syncope by haplology of "mononomial".

## 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 and second meaning.

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, in particular when monomial is used with the first meaning, 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.

The remainder of this article assumes the first meaning of "monomial".

## Monomial basis

Main article: Monomial basis

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, called the monomial basis - a fact of constant implicit use in mathematics.

## Number

The Hilbert series is a compact way to express the number of monomials of a given degree: the number of monomials of degree d in n variables is the coefficient of degree d of the formal power series expansion of

## Notation

we can define

for compactness.

## Degree

The degree of a monomial is sometimes called order, mainly in the context of series.

It is also called total degree when it is needed to distinguish it from the degree in one of the variables.

Monomial degree is fundamental to the theory of univariate and multivariate polynomials.

Explicitly, it is used to define the degree of a polynomial and the notion of homogeneous polynomial, as well as for graded monomial orderings used in formulating and computing Gröbner bases.

Implicitly, it is used in grouping the terms of a Taylor series in several variables.

## Geometry

## See also

- Monomial representation
- Monomial matrix
- Homogeneous polynomial
- Homogeneous function
- Multilinear form
- Log-log plot
- Power law

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Monomial.