The Laurent norm

TL;DR

This paper introduces a generalized Laurent semi-norm for Laurent polynomials, providing a decomposition formula, identifying essential variables, and establishing conditions under which it becomes a norm, extending concepts from 3-manifold topology.

Contribution

It generalizes the Laurent semi-norm from 3-manifold Alexander polynomials to arbitrary Laurent polynomials, with a decomposition formula and a reduced norm in essential variables.

Findings

Decomposition formula for Laurent norm

Identification of essential variables and reduced norm

Conditions under which the Laurent semi-norm becomes a norm

Abstract

We generalize a semi-norm for the Alexander polynomial of a connected, compact, oriented 3-manifold on its first cohomology group to a semi-norm 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 \leq 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.