Laurent polynomial
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Laurent polynomial (named after
Pierre Alphonse Laurent Pierre Alphonse Laurent (18 July 1813 – 2 September 1854) was a French mathematician, engineer, and Military Officer best known for discovering the Laurent series, an expansion of a function into an infinite power series, generalizing the Tayl ...
) in one variable over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
\mathbb is a linear combination of positive and negative powers of the variable with coefficients in \mathbb. Laurent polynomials in ''X'' form a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
denoted \mathbb , X^/math>. They differ from ordinary
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
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
complex variables Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebrai ...
.


Definition

A Laurent polynomial with coefficients in a field \mathbb is an expression of the form : p = \sum_k p_k X^k, \quad p_k \in \mathbb where ''X'' is a formal variable, the summation index ''k'' is an
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
(not necessarily positive) and only finitely many coefficients ''p''''k'' 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 ''X'' can be present: :\bigg(\sum_i a_i X^i\bigg) + \bigg(\sum_i b_i X^i\bigg) = \sum_i (a_i+b_i)X^i and :\bigg(\sum_i a_i X^i\bigg) \cdot \bigg(\sum_j b_j X^j\bigg) = \sum_k \Bigg(\sum_ a_i b_j\Bigg)X^k. Since only finitely many coefficients ''a''''i'' and ''b''''j'' are non-zero, all sums in effect have only finitely many terms, and hence represent Laurent polynomials.


Properties

* A Laurent polynomial over C may be viewed as a
Laurent series In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion c ...
in which only finitely many coefficients are non-zero. * The ring of Laurent polynomials ''R'' 'X'', ''X'' −1is an extension of the
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
''R'' 'X'' obtained by "inverting ''X'' ". More rigorously, it is the
localization Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is a ...
of the polynomial ring in the
multiplicative set In abstract algebra, a multiplicatively closed set (or multiplicative set) is a subset ''S'' of a ring ''R'' such that the following two conditions hold: * 1 \in S, * xy \in S for all x, y \in S. In other words, ''S'' is closed under taking finite ...
consisting of the non-negative powers of ''X''. 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 functions. * The ring of Laurent polynomials over a field is
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
(but not Artinian). * If ''R'' is an
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
, the
units Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * Unit (album), ...
of the Laurent polynomial ring ''R'' 'X'', ''X'' −1have the form ''uX'' ''k'', where ''u'' is a unit of ''R'' and ''k'' is an integer. In particular, if ''K'' is a field then the units of ''K'' 'X'', ''X'' −1have the form ''aX'' ''k'', where ''a'' is a non-zero element of ''K''. * The Laurent polynomial ring ''R'' 'X'', ''X'' −1is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to the
group ring In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the giv ...
of the
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
Z of
integers An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
over ''R''. More generally, the Laurent polynomial ring in ''n'' variables is isomorphic to the group ring of the
free abelian group In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subse ...
of rank ''n''. It follows that the Laurent polynomial ring can be endowed with a structure of a commutative,
cocommutative In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagram ...
Hopf algebra.


See also

*
Jones polynomial In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynom ...


References

* Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, , MR 1878556 {{DEFAULTSORT:Laurent Polynomial Commutative algebra Polynomials Ring theory