Laurent Polynomial

A Laurent polynomial with coefficients in the field F is an algebraic object that is typically expressed in the form


where the a_i are elements of F, and only finitely many of the a_i are nonzero. A Laurent polynomial is an algebraic object in the sense that it is treated as a polynomial except that the indeterminant "t" can also have negative powers.

Expressed more precisely, the collection of Laurent polynomials with coefficients in a field F form a ring, denoted F[t,t^(-1)], with ring operations given by componentwise addition and multiplication according to the relation


for all n and m in the integers. Formally, this is equivalent to saying that F[t,t^(-1)] is the group ring of the integers and the field F. This corresponds to F[t] (the polynomial ring in one variable for F) being the group ring or monoid ring for the monoid of natural numbers and the field F.

See also

Polynomial, Principal Part

Explore with Wolfram|Alpha


Lang, S. Undergraduate Algebra, 2nd ed. New York: Springer-Verlag, 1990.

Referenced on Wolfram|Alpha

Laurent Polynomial

Cite this as:

Weisstein, Eric W. "Laurent Polynomial." From MathWorld--A Wolfram Web Resource.

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