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In mathematics, a **Laurent polynomial** (named
after Pierre Alphonse Laurent) in one variable over a field is a linear combination of positive and negative powers of the variable with coefficients in . Laurent polynomials in form a ring denoted .^{[1]} They differ from ordinary polynomials 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.

A **Laurent polynomial** with coefficients in a field is an expression of the form

where is a formal variable, the summation index is an integer (not necessarily positive) and only finitely many coefficients 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 can be present:

and

Since only finitely many coefficients and are non-zero, all sums in effect have only finitely many terms, and hence represent Laurent polynomials.

- A Laurent polynomial over may be viewed as a Laurent series in which only finitely many coefficients are non-zero.
- The ring of Laurent polynomials is an extension of the polynomial ring obtained by "inverting ". More rigorously, it is the localization of the polynomial ring in the multiplicative set consisting of the non-negative powers of . 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 (but not Artinian).
- If is an integral domain, the units of the Laurent polynomial ring have the form , where is a unit of and is an integer. In particular, if is a field then the units of have the form , where is a non-zero element of .
- The Laurent polynomial ring is isomorphic to the group ring of the group of integers over . More generally, the Laurent polynomial ring in variables is isomorphic to the group ring of the free abelian group of rank . It follows that the Laurent polynomial ring can be endowed with a structure of a commutative, cocommutative Hopf algebra.

- Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556

**^**Weisstein, Eric W. "Laurent Polynomial".*MathWorld*.