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Laurent Polynomial

A Laurent polynomial with Coefficients in the Field $\Bbb{F}$ is an algebraic object that is typically expressed in the form

\ldots+ a_{-n}t^{-n} + a_{-(n-1)}t^{-(n-1)} + \ldots + a_{-1}t^{-1} + a_0 + a_1t + \ldots + a_n t^n + \ldots,

where the $a_i$ are elements of $\Bbb{F}$, 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 $\Bbb{F}$ form a Ring, denoted $\Bbb{F}[t,t^{-1}]$, with Ring operations given by componentwise addition and multiplication according to the relation

at^n \cdot bt^m = ab t^{n+m}

for all $n$ and $m$ in the Integers. Formally, this is equivalent to saying that $\Bbb{F}[t,t^{-1}]$ is the Group Ring of the Integers and the Field $\Bbb{F}$. This corresponds to $\Bbb{F}[t]$ (the Polynomial ring in one variable for $\Bbb{F}$) being the Group Ring or Monoid ring for the Monoid of natural numbers and the Field $\Bbb{F}$.

See also Polynomial


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

© 1996-9 Eric W. Weisstein