Infinite Minkowski sums of lattice polyhedra

Abstract

Artinian integrally closed monomial ideals are characterized by their Newton polyhedra, which are lattice polyhedra inside the positive orthant having the positive orthant as their recession cone. Multiplication of such ideals correspond to Minkowski addition of their Newton polyhedra. In two dimensions, the isomorphic monoids of artinian, integrally closed monomial ideals under multiplication, or the class of lattice polyhedra described above, under Minkowski addition, are free abelian, as proved by Crispin-Quinonez. Bayer and Stillman considered so-called /monomial submodules/ of the Laurent polynomial ring. Inspired by this, we consider a family of such monomial submodules that can be (uniquely) expressed as an infinite product of monomial submodules isomorphic to integrally closed monomial ideals. Geometrically, their Newton polyhedras are expressed as an infinite Minkowski sum of *simple* lattice polyhedra. This gives another example of a *topological ufd* i.e. a topological abelian monoid in which every element can be uniquely written as a convergent (possibly infinite) product of irreducibles.