Laurent polynomial

A Laurent polynomial permits finitely many positive and negative integer powers, so it has the form

\sum_{k = - m}^{n} a_{k} z^{k}

Table of contents

Breadth of a Laurent polynomial ( $br f = max supp f - min supp f$ )

 0  0

For a nonzero Laurent polynomial, its breadth is the largest exponent with nonzero coefficient minus the smallest such exponent. Unlike its highest exponent, it is unchanged by multiplying by a Laurent monomial unit. The breadth of a nonzero constant is zero.

Laurent monomial ( $c \prod_{i} X_{i}^{a_{i}}, a_{i} \in Z$ )

 0  0

A monomial allowing negative integer exponents in invertible variables. Finite sums of Laurent monomials form a Laurent polynomial ring. In a graded coordinate ring each has total degree

\sum_{i} a_{i}

Constant term ( $CT$ )

 1  0

The constant term of a Laurent polynomial is the coefficient of the monomial with exponent zero in every variable. Taking the constant term in just one variable leaves a Laurent polynomial in the other variables; successive extractions recover the full constant term. This operation is linear, but need not preserve products. The Dyson constant-term identity is a multivariate example of coefficient extraction.