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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:
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
*For the first definition, a '''monomial''' is a product of powers of [[Variable (mathematics)|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 {{mvar|x}}<sup>0</sup> for any variable {{mvar|x}}. 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</math>, <math>y</math>, <math>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 (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. A monomial in the first sense is also a monomial in the second sense, because the multiplication by 1 is allowed. For example, <math>-7x^5</math> and <math>(3-4i)x^4yz^{13}</math> are monomials (in the second example, the variables are <math>x</math>, <math>y</math>, <math>z</math>, and the constant coefficients are [[complex numbers]]).


In the context of [[Laurent polynomial]]s and [[Laurent series]], the exponents of a '''monomial''' may be negative, and in the context of [[Puiseux series]], the exponents may be [[rational number]]s.
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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.


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 = | coauthors = John Little, 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, and an [http://planetmath.org/encyclopedia/Monomial.html 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 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, 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.
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


''The remainder of this article assumes the first meaning of "monomial".''
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


==As bases==
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


The most obvious fact about monomials (first meaning) is that any polynomial is a [[linear combination]] of them, so they form a [[basis (linear algebra)|basis]] of the [[vector space]] of all polynomials - a fact of constant implicit use in mathematics.
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


==Number==
==Demos==
The number of monomials of degree ''d'' in ''n'' variables is the number of [[multicombination]]s 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]] <math>\textstyle{\left(\!\!{n\choose d}\!\!\right)}</math>. This expression can also be given in the form of a [[binomial coefficient]], as a [[polynomial expression]] in ''d'', or using a [[Pochhammer symbol#Alternate notations|rising factorial power]] of {{nowrap|''d'' + 1}}:
:<math>\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}}.</math>
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 <math>n-1</math> with leading coefficient <math>\tfrac1{(n-1)!}</math>.


For example, the number of monomials in three variables (<math>n=3</math>) of degree ''d'' is <math>\textstyle{\frac{1}{2}}(d+1)^{\overline2} = \textstyle{\frac{1}{2}}(d+1)(d+2)</math>; these numbers form the sequence 1, 3, 6, 10, 15, ... of [[triangular number]]s.
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


The [[Hilbert series]] is a compact way to express the number of monomials of a given degree: the number of monomials of degree {{mvar|d}} in {{mvar|n}} variables is the coefficient of degree {{mvar|d}} of the [[formal power series]] expansion of
:<math> \frac{1}{(1-t)^n}.</math>


==Notation==
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


Notation for monomials is constantly required in fields like [[partial differential equation]]s. If the variables being used form an indexed family like <math>x_1</math>, <math>x_2</math>, <math>x_3</math>, ..., then ''[[multi-index notation]]'' is helpful: if we write
==Test pages ==


:<math>\alpha = (a, b, c)</math>
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


we can define
*[[Inputtypes|Inputtypes (private Wikis only)]]
:<math>x^{\alpha} = x_1^a\, x_2^b\, x_3^c</math>
*[[Url2Image|Url2Image (private Wikis only)]]
 
==Bug reporting==
and save a great deal of space.
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
 
==Degree<!-- [[Degree of a monomial]] redirects here -->==
 
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 <math>a+b+c</math>. The degree of <math>x y z^2</math> is 1+1+2=4.
 
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 ordering]]s used in formulating and computing [[Gröbner basis|Gröbner bases]]. Implicitly, it is used in grouping the terms of a [[Taylor series#Taylor series in several variables|Taylor series in several variables]].
 
==Geometry==
 
In [[algebraic geometry]] the varieties defined by monomial equations <math>x^{\alpha} = 0</math> for some set of α have special properties of homogeneity. This can be phrased in the language of [[algebraic group]]s, in terms of the existence of a [[group action]] of an [[algebraic torus]] (equivalently by a multiplicative group of [[diagonal matrix|diagonal matrices]]). This area is studied under the name of ''[[Toric geometry|torus embedding]]s''.
 
==See also==
* [[Monomial representation]]
* [[Generalized permutation matrix|Monomial matrix]]
* [[Homogeneous polynomial]]
* [[Homogeneous function]]
* [[Multilinear form]]
* [[Log-log plot]]
* [[Power law]]
 
== Notes ==
 
{{Reflist}}
 
[[Category:Homogeneous polynomials]]
[[Category:Algebra]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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