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.In mathematical analysis, it is common to consider polynomials written in terms of a shifted variable[3][4] By a slight abuse of notation, monomials of shifted variables, for instanceSince 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".[5] 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[6] and second[7] 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.An argument in favor of the first meaning is that no obvious other notion is available to designate these values,[citation needed] though primitive monomial is in use and does make the absence of constants clear.[1] The remainder of this article assumes the first meaning of "monomial".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.: The latter forms are particularly useful when one fixes the number of variables and lets the degree vary.The Hilbert series is a compact way to express the number of monomials of a given degree: the number of monomials of degreeof the formal power series expansion of The number of monomials of degree at most d in n variables isThis follows from the one-to-one correspondence between the monomials of degreeIf the variables being used form an indexed family likeone 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: The degree of a monomial is defined as the sum of all the exponents of the variables, including the implicit exponents of 1 for the variables which appear without exponent; e.g., in the example of the previous section, the degree isThe degree of a monomial is sometimes called order, mainly in the context of series.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.In algebraic geometry the varieties defined by monomial equationsfor some set of α have special properties of homogeneity.