In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:

1. A monomial, also called a power product or primitive monomial, is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. is a primitive monomial, being equal to the empty product).

2. A monomial in the first sense is multiplied by a nonzero constant, called the coefficient of the monomial. and the coefficient is a complex number).

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.

In mathematical analysis alone, as in the study of Taylor series. By a slight abuse of notation may be called monomials in the sense of shifted monomials or centered monomials is the center is the shift.

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 that no obvious other notion is available to designate these values, though primitive monomial is in use and does make the absence of constants clear.

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

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

variables is the number of multicombinations variables (a variable can be chosen more than once, but order does not matter), which is given by the multiset coefficient. This expression can also be given in the form of a binomial coefficient, as a polynomial expression, or using a rising factorial power:

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.

) of degree d; these numbers form the sequence 1, 3, 6, 10, 15, ... of triangular numbers.

The Hilbert series of the formal power series expansion of

The number of monomials of degree at most in variables, which consists in substituting by 1 the extra variable.

Multi-index notation

The multi-index notation one can set

and

Then the monomial

can be compactly written as

With this notation, the product of two monomials is simply expressed by using the addition of exponent vectors:

Degree

is 1+1+2=4. The degree of a nonzero constant is 0. For example, the degree of −7 is 0.

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

In algebraic geometry 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 embeddingss.

See also

• Monomial representation

• Monomial matrix

• Homogeneous polynomial

• Homogeneous function

• Multilinear form

• Log-log plot

• Power law

• Sparse polynomial

References