Editing Laurent polynomial

{{Short description|Polynomial with finitely many terms of the form axⁿ where n ∈ Z}}
{{More citations needed|date=July 2009}}

In [[mathematics]], a '''Laurent polynomial''' (named
after [[Pierre Alphonse Laurent]]) in one variable over a [[Field (mathematics)|field]] <math>\mathbb{F}</math> is a [[linear combination]] of positive and negative powers of the variable with [[coefficient]]s in <math>\mathbb{F}</math>. Laurent polynomials in <math>X</math> form a [[Ring (mathematics)|ring]] denoted <math>\mathbb{F}[X, X^{-1}]</math>.<ref>{{MathWorld|urlname=LaurentPolynomial|title=Laurent Polynomial}}</ref> They differ from ordinary [[polynomial]]s in that they may have terms of negative degree. The construction of Laurent polynomials may be iterated, leading to the ring of Laurent polynomials in several variables.  Laurent polynomials are of particular importance in the study of [[several complex variables|complex variables]].

== Definition ==
A '''Laurent polynomial''' with coefficients in a field <math>\mathbb{F}</math> is an expression of the form

: <math>p = \sum_k p_k X^k, \quad p_k \in \mathbb{F}</math>

where <math>X</math> is a formal variable, the summation index <math>k</math> is an [[integer]] (not necessarily positive) and only finitely many coefficients <math>p_{k}</math> are non-zero. Two Laurent polynomials are equal if their coefficients are equal. Such expressions can be added, multiplied, and brought back to the same form by reducing similar terms. Formulas for addition and multiplication are exactly the same as for the ordinary polynomials, with the only difference that both positive and negative powers of <math>X</math> can be present:

:<math>\bigg(\sum_i a_i X^i\bigg) + \bigg(\sum_i b_i X^i\bigg) = 
\sum_i (a_i+b_i)X^i</math>
and
:<math>\bigg(\sum_i a_i X^i\bigg) \cdot \bigg(\sum_j b_j X^j\bigg) = 
\sum_k \Bigg(\sum_{i,j \atop i+j=k} a_i b_j\Bigg)X^k.</math>

Since only finitely many coefficients <math>a_{i}</math> and <math>b_{j}</math> are non-zero, all sums in effect have only finitely many terms, and hence represent Laurent polynomials.

== Properties ==

* A Laurent polynomial over <math>\mathbb{C}</math> may be viewed as a [[Laurent series]] in which only finitely many coefficients are non-zero.
* The ring of Laurent polynomials <math>R\left [X, X^{-1} \right ]</math> is an extension of the [[polynomial ring]] <math>R[X]</math> obtained by "inverting <math>X</math>". More rigorously, it is the [[localization of a ring|localization]] of the polynomial ring in the [[multiplicative set]] consisting of the non-negative powers of <math>X</math>. Many properties of the Laurent polynomial ring follow from the general properties of localization.
* The ring of Laurent polynomials is a [[subring]] of the [[rational function]]s.
* The ring of Laurent polynomials over a field is [[Noetherian ring|Noetherian]] (but not [[Artinian ring|Artinian]]).
* If <math>R</math> is an [[integral domain]], the [[unit (ring theory)|units]] of the Laurent polynomial ring <math>R\left [X, X^{-1} \right ]</math> have the form <math>uX^{k}</math>, where <math>u</math> is a unit of <math>R</math> and <math>k</math> is an integer. In particular, if <math>K</math> is a field then the units of <math>K[X, X^{-1}]</math> have the form <math>aX^{k}</math>, where <math>a</math> is a non-zero element of <math>K</math>.
* The Laurent polynomial ring <math>R[X, X^{-1}]</math> is [[isomorphic]] to the [[group ring]] of the [[group (mathematics)|group]] <math>\mathbb{Z}</math> of [[Integer#Algebraic properties|integers]] over <math>R</math>. More generally, the Laurent polynomial ring in <math>n</math> variables is isomorphic to the group ring of the [[free abelian group]] of rank <math>n</math>. It follows that the Laurent polynomial ring can be endowed with a structure of a commutative, [[cocommutative]] [[Hopf algebra]].

==See also==
*[[Jones polynomial]]

==References==
{{Reflist}}

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{{DEFAULTSORT:Laurent Polynomial}}
[[Category:Commutative algebra]]
[[Category:Polynomials]]
[[Category:Ring theory]]