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Infinite Minkowski sums of lattice polyhedra

Jan Snellman

TL;DR

This paper investigates infinite factorization of a restricted class of integrally closed monomial modules in two variables by translating algebraic questions into convex-geometric ones via Newton polyhedra and Minkowski sums. It constructs a topological unique factorization monoid (topological ufd) by representing elements as convergent infinite sums of simple pieces, formalized through an isomorphism $\eta: \mathcal{M} \to \mathcal{A}$ and a convergent Minkowski-sum decomposition $\mathrm{New}(M) = \sum_{j\ge1} N(F_j)$. A concrete model is given where $\mathcal{M} = \mathbb{N}^{\mathbb{N}_+}$ parameterizes infinite factorizations, with $N(F_j) = \mathbb{R}_+^{2} + \ell_j$ and $\ell_j$ the segment to $(j,-1)$, yielding a unique infinite product representation of integrally closed monomial modules. The results illuminate a bridge between combinatorial semigroup theory, convex geometry, and commutative algebra, and point toward extensions and challenges in higher dimensions where unique factorization may fail.

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.

Infinite Minkowski sums of lattice polyhedra

TL;DR

This paper investigates infinite factorization of a restricted class of integrally closed monomial modules in two variables by translating algebraic questions into convex-geometric ones via Newton polyhedra and Minkowski sums. It constructs a topological unique factorization monoid (topological ufd) by representing elements as convergent infinite sums of simple pieces, formalized through an isomorphism and a convergent Minkowski-sum decomposition . A concrete model is given where parameterizes infinite factorizations, with and the segment to , yielding a unique infinite product representation of integrally closed monomial modules. The results illuminate a bridge between combinatorial semigroup theory, convex geometry, and commutative algebra, and point toward extensions and challenges in higher dimensions where unique factorization may fail.

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.

Paper Structure

This paper contains 6 sections, 11 theorems, 40 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Monomial modules and monoid submodules
  3. Factorization of integrally closed monomial ideals
  4. Infinite factorization
  5. Further questions
  6. Bibliography

Key Result

Proposition 6

Addition and inversion in $\mathcal{M}$ are continuous, i.e., $\mathcal{M}$ is a topological monoid. Furthermore, $\mathcal{M}$ is a topological unique factorization monoid, and it is the inverse limit of the surjective inverse system where each is a discrete, finite rank free abelian monoid, and the restriction maps are

Figures (7)

  • Figure 1: An infinite power of the maximal ideal
  • Figure 2: The Newton polyhedra of a non-finitely generated integrally closed monomial module, expressed as an infinite Minkowski sum of simple polyhedra.
  • Figure 3: A co-artinian monomial module
  • Figure 6: Ways to reach $(a,-3)$ with $a \le 6$.
  • Figure 7: $(2,2) = (0,2)+N(2F_{1})$ as the Minkowski sum of line segment and ${\mathbb{R}}^{2}_{+}$
  • ...and 2 more figures

Theorems & Definitions (32)

  • Example 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Definition 5
  • Proposition 6
  • proof
  • Definition 7
  • Example 8
  • Lemma 9
  • ...and 22 more