The Laurent Norm

We generalize a semi-norm for the Alexander polynomial of a connected, compact, oriented 3-manifold on its first cohomology group to a seminorm for an arbitrary Laurent polynomial f on the dual vector space to the space of exponents of f . We determine a decomposition formula for this Laurent norm; an expression for the Laurent norm for f in terms of the Laurent norms for each of the irreducible factors of f . For an n-variable polynomial f , we introduce a space of m ≤ n essential variables which determine the reduced Laurent norm unit ball; a convex polyhedron of the same dimension m as the Newton polyhedron of f . In the space spanned by the essential variables, the Laurent semi-norm for polynomials with at least two terms is shown to be a norm.