Editing Laurent polynomial (section)

== Definition ==
A '''Laurent polynomial''' with coefficients in a field <math>\mathbb{F}</math> is an expression of the form

: <math>p = \sum_k p_k X^k, \quad p_k \in \mathbb{F}</math>

where <math>X</math> is a formal variable, the summation index <math>k</math> is an [[integer]] (not necessarily positive) and only finitely many coefficients <math>p_{k}</math> 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 <math>X</math> can be present:

:<math>\bigg(\sum_i a_i X^i\bigg) + \bigg(\sum_i b_i X^i\bigg) = 
\sum_i (a_i+b_i)X^i</math>
and
:<math>\bigg(\sum_i a_i X^i\bigg) \cdot \bigg(\sum_j b_j X^j\bigg) = 
\sum_k \Bigg(\sum_{i,j \atop i+j=k} a_i b_j\Bigg)X^k.</math>

Since only finitely many coefficients <math>a_{i}</math> and <math>b_{j}</math> are non-zero, all sums in effect have only finitely many terms, and hence represent Laurent polynomials.