A Laurent

polynomial in

*x* (over a

field *k*)
is an expression of the form

*a*_{-n}x^{-n}+...+a_{0}+a_{1}x+...+a_{m}x^{m}
where the

coefficients

*a*_{i} lie in

*k*. (You can take

*k* to be the

real numbers or

complex numbers.)

For example, *10+5x*, *x*^{-1}, and *x*^{-2}+3x^{2}
are all Laurent polynomials.

They can be added and multiplied in just the way you expect (so that *x.x*^{-1}=1)
and the collection *k[x,x*^{-1}] of all Laurent polynomials
forms a commutative ring.