Are Laurent series called polynomials?

Question

Traditionally, polynomials cannot have negative exponents. So what gives? Inspired by this.

Answer 1

By definition, a polynomial has only nonnegative powers of its variables, while Laurent series have some powers of negative degree. Thus Laurent series and polynomials are disjoint.

If, however, there are only finitely many positive and negative powers in a Laurent series, it may be called a Laurent polynomial.

Answer 2

A Laurent polynomial is an expression of the form $p(x,x^{-1})$, where $p$ is a polynomial in two variables.

A Laurent polynomial is a Laurent series with only finitely many terms.