Monomial

Monomial

In mathematics, in the context of polynomials, the word monomial can have one of two different meanings:

  • The first is a product of powers of variables, or formally any value obtained by finitely many multiplications of a variable. If only a single variable x is considered, this means that any monomial is either 1 or a power xn of x, with n a positive integer. If several variables are considered, say, x, y, z, then each can be given an exponent, so that any monomial is of the form xaybzc with a,b,c non-negative integers (taking note that any exponent 0 makes the corresponding factor equal to 1).
  • The second meaning of monomial includes monomials in the first sense, but also allows multiplication by any constant, so that − 7x5 and (3 − 4i)x4yz13 are also considered to be monomials (the second example assuming polynomials in x, y, z over the complex numbers are considered).

Contents

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[1] and second[2] 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, 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. For an isolated polynomial consisting of a single term, one could if necessary use the uncontracted form mononomial, analogous to binomial and trinomial.

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

As bases

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 - a fact of constant implicit use in mathematics.

Number

The number of monomials of degree d in n variables is the number of multicombinations of d elements chosen among the n variables (a variable can be chosen more than once, but order does not matter), which is given by the multiset coefficient \textstyle{\left(\!\!{n\choose d}\!\!\right)}. This expression can also be given in the form of a binomial coefficient, as a polynomial expression in d, or using a rising factorial power of d + 1:

\left(\!\!{n\choose d}\!\!\right) = \binom{n+d-1}{d} = \binom{d+(n-1)}{n-1}
  = \frac{(d+1)\times(d+2)\times\cdots\times(d+n-1)}{1\times2\times\cdots\times(n-1)} = \frac{1}{(n-1)!}(d+1)^{\overline{n-1}}.

The latter forms are particularly useful when one fixes the number of variables and lets the degree vary. From these expressions one sees that for fixed n, the number of monomials of degree d is a polynomial expression in d of degree n − 1 with leading coefficient \tfrac1{(n-1)!}.

For example, the number of monomials in three variables (n = 3) of degree d is \textstyle{\frac{1}{2}}(d+1)^{\overline2} = \textstyle{\frac{1}{2}}(d+1)(d+2); these numbers form the sequence 1, 3, 6, 10, 15, ... of triangular numbers.

Notation

Notation for monomials is constantly required in fields like partial differential equations. If the variables being used form an indexed family like x1, x2, x3, ..., then multi-index notation is helpful: if we write

α = (a,b,c)

we can define

x^{\alpha} = x_1^a\, x_2^b\, x_3^c

and save a great deal of space.

Geometry

In algebraic geometry the varieties defined by monomial equations xα = 0 for some set of α have special properties of homogeneity. This can be phrased in the language of algebraic groups, in terms of the existence of a group action of an algebraic torus (equivalently by a multiplicative group of diagonal matrices). This area is studied under the name of torus embeddings.

See also

Notes

  1. ^ Cox, David; John Little, Donal O'Shea (1998). Using Algebraic Geometry. Springer Verlag. pp. 1. ISBN 0-387-98487-9. 
  2. ^ Hazewinkel, Michiel, ed. (2001), "Monomial", Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/M/m064760.htm 

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  • monomial — [mō nō′mē əl, mänō′mē əl] adj. [ MO(NO) + (BI)NOMIAL] 1. Algebra consisting of only one term 2. Biol. consisting of only one word: said of a taxonomic name n. a monomial expression, quantity, or name …   English World dictionary

  • Monomial — Mo*no mi*al, n. [See {Monome}, {Binomial}.] (Alg.) A single algebraic expression; that is, an expression unconnected with any other by the sign of addition, substraction, equality, or inequality. [1913 Webster] …   The Collaborative International Dictionary of English

  • Monomial — Mo*no mi*al, a. (Alg.) Consisting of but a single term or expression. [1913 Webster] …   The Collaborative International Dictionary of English

  • monomial — 1. adjective Relative to an algebraic expression consisting of one term. 2. noun a) A single term consisting of a product of variables with positive integer exponents. x²y is a monomial b) A single term consisti …   Wiktionary

  • monomial — noun Etymology: blend of mon and nomial (as in binomial) Date: circa 1706 1. a mathematical expression consisting of a single term 2. a taxonomic name consisting of a single word or term • monomial adjective …   New Collegiate Dictionary

  • monomial — /moh noh mee euhl, meuh /, adj. 1. Algebra. a. consisting of one term only. b. (of a matrix) having exactly one non zero term in each row and each column. 2. Biol. noting or pertaining to a name that consists of a single word or term. n. 3.… …   Universalium

  • monomial — mo·no·mi·al || mÉ™ nəʊmɪəl n. one term or letter (Algebra) adj. made of one term; of a monomial (Algebra) …   English contemporary dictionary

  • monomial — /mɒˈnoʊmiəl/ (say mo nohmeeuhl) Algebra –adjective 1. consisting of one term only. –noun 2. a monomial expression or quantity. {irregularly from mo(no) + nomial, after binomial} …  

  • monomial — adj. & n. Math. adj. (of an algebraic expression) consisting of one term. n. a monomial expression. Etymology: MONO after binomial …   Useful english dictionary

  • monomial Mathematics — [mə nəʊmɪəl] adjective (of an algebraic expression) consisting of one term. noun a monomial expression. Origin C18: from mono , on the pattern of binomial …   English new terms dictionary

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