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Concept# Monomial

Summary

In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
A monomial, also called power product, is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. For example, is a monomial. The constant is a monomial, being equal to the empty product and to for any variable . If only a single variable is considered, this means that a monomial is either or a power of , with a positive integer. If several variables are considered, say, then each can be given an exponent, so that any monomial is of the form with non-negative integers (taking note that any exponent makes the corresponding factor equal to ).
A monomial is a monomial in the first sense multiplied by a nonzero constant, called the coefficient of the monomial. A monomial in the first sense is a special case of a monomial in the second sense, where the coefficient is . For example, in this interpretation and are monomials (in the second example, the variables are 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.
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".
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.

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