Laurent polynomial

In mathematics, a Laurent polynomial (named
after Pierre Alphonse Laurent) in one variable over a field formula_1 is a linear combination of positive and negative powers of the variable with coefficients in formula_1. Laurent polynomials in "X" form a ring denoted formula_1["X", "X"−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.
Definition.
A Laurent polynomial with coefficients in a field formula_1 is an expression of the form
where "X" is a formal variable, the summation index "k" is an integer (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:
and
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.