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In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two different definitions of a monomial may be encountered: *For the first definition, 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. The constant 1 is a monomial, being equal to the empty product and 0 for any variable . If only a single variable is considered, this means that a monomial is either 1 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 0 makes the corresponding factor equal to 1). *For the second definition, 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 also a monomial in the second sense, because the multiplication by 1 is allowed. 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 "polynomial" comes from "poly-" plus the Greek word "νομός" (nomós, meaning part, portion), a monomial should theoretically be called a "mononomial". "Monomial" is a syncope of "mononomial".〔''American Heritage Dictionary of the English Language'', 1969.〕 == 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, and an (unclear definition ). 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 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. ''The remainder of this article assumes the first meaning of "monomial".'' 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「monomial」の詳細全文を読む スポンサード リンク
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