Masonbiamonte (talk | contribs) m The coefficient was described as an "imaginary complex number" but the coefficient indeed has a non-zero real component and so is a general complex number. Tag: Visual edit |
The enemies of god (talk | contribs) revised florid constructions, ambiguous and conflating statements, and removed modal assertions about the etymology of the word |
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In [[mathematics]] |
In [[mathematics]] a '''monomial''' is a simple product. It's a product of powers (a "''power product''") of [[Variable (mathematics)|variables]] with [[nonnegative integer]] exponents, possibly multiplied by a nonzero constant called the [[coefficient]] of the monomial. |
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*(2): 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 a special case of a monomial in the second sense, where the coefficient is 1. For example, in this interpretation <math>-7x^5</math> and <math>(3-4i)x^4yz^{13}</math> are monomials (in the second example, the variables are <math>x, y, z,</math> and the coefficient is a [[complex number]]). |
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Trivially, a monomial is a single [https://www.mathsisfun.com/definitions/term.html term] of a [https://www.mathsisfun.com/algebra/polynomials.html polynomial], or a [[polynomial]] of only one [[term]]. The set of monomials is a subset of all polynomials that is closed under multiplication, and monomials are combined into polynomials by addition and subtraction, at which point they are called ''terms'' of the polynomial. Since both single constants and single variables can be atomic terms of a polynomial, ''term'' and ''monomial'' become synonymous in that usage, provided the restrictions of ''[[polynomial]]'' are observed. |
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The single variable case is easy to see as a lone variable has both the implicit exponent of 1 and the implicit coefficient of 1, and so meets the definition of a product of powers with nonnegative integer exponent and multiplied by a nonzero constant. |
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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 (phonetics)|syncope]] by [[haplology]] of "mononomial".<ref>''American Heritage Dictionary of the English Language'', 1969.</ref> |
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For example, <math>x</math> or <math>1x^1</math>, 6, <math>-7x^5</math> and <math>(3-4i)x^4yz^{13}</math> are monomials (in the fourth example the variables are <math>x, y, z,</math> and the coefficient is a [[complex number]]). The constant 1 is also a monomial by virtue of being equal to the [[empty product]] and {{mvar|x}}<sup>0</sup> for any variable {{mvar|x}}. |
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== Comparison of the two definitions == |
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With either definition, the set of monomials is a subset of all polynomials that is closed under multiplication. |
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⚫ | If only a single variable {{mvar|x}} is considered, this means that a monomial is either 1 or a power {{math|''x''<sup>''n''</sup>}} of {{mvar|x}}, with {{mvar|n}} a positive integer. If several variables are considered, say, <math>x, y, z,</math> then each can be given an exponent, so that any monomial is of the form <math>x^a y^b z^c</math> with <math>a,b,c</math> non-negative integers. |
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Both uses of this notion can be found, and in many cases the distinction is simply ignored, see for instance examples for the first<ref>{{cite book | last = Cox | first = David | authorlink = |author2=John Little |author3=Donal O'Shea | title = Using Algebraic Geometry | publisher = Springer Verlag | year = 1998 | location = | pages = 1 | url = | doi = | id = | isbn = 0-387-98487-9 }}</ref> and second<ref>{{Springer|id=M/m064760|title=Monomial}}</ref> 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 order]]ing 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. |
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''The remainder of this article assumes the first meaning of "monomial".'' |
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==Monomial basis== |
==Monomial basis== |
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{{main|Monomial basis}} |
{{main|Monomial basis}} |
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Every polynomial is a [[linear combination]] of monomials; therefore the monomials form a [[basis (linear algebra)|basis]] of the [[vector space]] of all polynomials, the [[monomial basis]]. |
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==Number== |
==Number== |
Revision as of 22:37, 16 January 2019
In mathematics a monomial is a simple product. It's a product of powers (a "power product") of variables with nonnegative integer exponents, possibly multiplied by a nonzero constant called the coefficient of the monomial.
Trivially, a monomial is a single term of a polynomial, or a polynomial of only one term. The set of monomials is a subset of all polynomials that is closed under multiplication, and monomials are combined into polynomials by addition and subtraction, at which point they are called terms of the polynomial. Since both single constants and single variables can be atomic terms of a polynomial, term and monomial become synonymous in that usage, provided the restrictions of polynomial are observed.
The single variable case is easy to see as a lone variable has both the implicit exponent of 1 and the implicit coefficient of 1, and so meets the definition of a product of powers with nonnegative integer exponent and multiplied by a nonzero constant.
For example, or , 6, and are monomials (in the fourth example the variables are and the coefficient is a complex number). The constant 1 is also a monomial by virtue of being equal to the empty product and x0 for any variable x.
If only a single variable x is considered, this means that a monomial is either 1 or a power xn of x, with n 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.
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 basis
Every polynomial is a linear combination of monomials; therefore the monomials form a basis of the vector space of all polynomials, the monomial basis.
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 . 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:
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 with leading coefficient .
For example, the number of monomials in three variables () of degree d is ; these numbers form the sequence 1, 3, 6, 10, 15, ... of triangular numbers.
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
The number of monomials of degree at most d in n variables is This follows from the one-to-one correspondence between the monomials of degree d in n+1 variables and the monomials of degree at most d in n variables, which consists in substituting by 1 the extra variable.
Notation
Notation for monomials is constantly required in fields like partial differential equations. If the variables being used form an indexed family like , , , ..., then multi-index notation is helpful: if we write
we can define
and save a great deal of space.
Degree
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 is . The degree of is 1+1+2=4. The degree of a nonzero constant is 0. For example, the degree of -7 is 0.
The degree of a monomial is sometimes called order, mainly in the context of series. It is also called total degree when it is needed to distinguish it from the degree in one of the variables.
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.
Geometry
In algebraic geometry the varieties defined by monomial equations 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
- Monomial representation
- Monomial matrix
- Homogeneous polynomial
- Homogeneous function
- Multilinear form
- Log-log plot
- Power law