Monomial

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In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Monomial_sentence_0

Two definitions of a monomial may be encountered: Monomial_sentence_1

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. Monomial_sentence_2

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_sentence_3

"Monomial" is a syncope by haplology of "mononomial". Monomial_sentence_4

Comparison of the two definitions Monomial_section_0

With either definition, the set of monomials is a subset of all polynomials that is closed under multiplication. Monomial_sentence_5

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. Monomial_sentence_6

In informal discussions the distinction is seldom important, and tendency is towards the broader second meaning. Monomial_sentence_7

When studying the structure of polynomials however, one often definitely needs a notion with the first meaning. Monomial_sentence_8

This is for instance the case when considering a monomial basis of a polynomial ring, or a monomial ordering of that basis. Monomial_sentence_9

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. Monomial_sentence_10

The remainder of this article assumes the first meaning of "monomial". Monomial_sentence_11

Monomial basis Monomial_section_1

Main article: Monomial basis Monomial_sentence_12

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. Monomial_sentence_13

Number Monomial_section_2

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 Monomial_sentence_14

Notation Monomial_section_3

we can define Monomial_sentence_15

for compactness. Monomial_sentence_16

Degree Monomial_section_4

The degree of a monomial is sometimes called order, mainly in the context of series. Monomial_sentence_17

It is also called total degree when it is needed to distinguish it from the degree in one of the variables. Monomial_sentence_18

Monomial degree is fundamental to the theory of univariate and multivariate polynomials. Monomial_sentence_19

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. Monomial_sentence_20

Implicitly, it is used in grouping the terms of a Taylor series in several variables. Monomial_sentence_21

Geometry Monomial_section_5

See also Monomial_section_6

Monomial_unordered_list_0


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