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 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. For example, x 2 y z 3 = x x y z z z {\displaystyle x^{2}yz^{3}=xxyzzz} is a monomial. The constant 1 {\displaystyle 1} is a primitive monomial, being equal to the empty product and to x 0 {\displaystyle x^{0}} any variable x {\displaystyle x} . If only a single variable x {\displaystyle x} is considered, this means that a monomial is either 1 {\displaystyle 1} or a power x n {\displaystyle x^{n}} of x {\displaystyle x} , with n {\displaystyle n} a positive integer. If several variables are considered, say, x , y , z , {\displaystyle x,y,z,} then each can be given an exponent so that any monomial is of the form x a y b z c {\displaystyle x^{a}y^{b}z^{c}} with a , b , c {\displaystyle a,b,c} non-negative integers (taking note that any exponent 0 {\displaystyle 0} makes the corresponding factor equal to 1 {\displaystyle 1} ). A monomial in the first sense is multiplied by a nonzero constant, called the coefficient of the monomial. A primitive monomial is a special case of a monomial in this second sense, where the coefficient is 1 {\displaystyle 1} . For example, in this interpretation, − 7 x 5 {\displaystyle -7x^{5}} and ( 3 − 4 i ) x 4 y z 13 {\displaystyle (3-4i)x^{4}yz^{13}} are monomials (in the second example, the variables are x , y , z , {\displaystyle x,y,z,} 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, it is common to consider polynomials written in terms of a shifted variable x ¯ = x − c {\displaystyle {\bar {x}}=x-c} for some constant c {\displaystyle c} rather than a variable x {\displaystyle x} alone, as in the study of Taylor series. By a slight abuse of notation, monomials of shifted variables, for instance 2 x ¯ 3 = 2 ( x − c ) 3 , {\displaystyle 2{\bar {x}}^{3}=2(x-c)^{3},} may be called monomials in the sense of shifted monomials or centered monomials, where c {\displaystyle c} is the center or − c {\displaystyle -c} 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.
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