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Newton Polytopes and Analytic Spread

Benjamin Drabkin, Benjamin Oltsik

TL;DR

The paper develops a convex-geometric framework to determine the analytic spread $\ell(I)$ of monomial ideals using Newton polytopes and polyhedra. It derives a practical bound $\ell(I)\le n+1-(s+k)$ from facet data and provides a sharp criterion for when a monomial ideal is basic, linking generator count to hyperplane structure. The authors prove equality of the bound in several key settings, including three-dimensional Newton polytopes, ideals with disjointly generated primary decompositions, and intersections of two monomial primes, and they compute $\ell(I)$ for several families. They also raise open questions about the existence and nature of monomial reductions, offering a convex-geometric perspective on reductions and their generator-minimization properties.

Abstract

Using the Newton polytope and polyhedron, we study analytic spread and ideal reductions of monomial ideals. We determine a bound for analytic spread based on halfspaces and hyperplanes of the Newton polytope, and we classify basic monomial ideals. We then apply this method to calculate the analytic spread for a few families of monomial ideals.

Newton Polytopes and Analytic Spread

TL;DR

The paper develops a convex-geometric framework to determine the analytic spread of monomial ideals using Newton polytopes and polyhedra. It derives a practical bound from facet data and provides a sharp criterion for when a monomial ideal is basic, linking generator count to hyperplane structure. The authors prove equality of the bound in several key settings, including three-dimensional Newton polytopes, ideals with disjointly generated primary decompositions, and intersections of two monomial primes, and they compute for several families. They also raise open questions about the existence and nature of monomial reductions, offering a convex-geometric perspective on reductions and their generator-minimization properties.

Abstract

Using the Newton polytope and polyhedron, we study analytic spread and ideal reductions of monomial ideals. We determine a bound for analytic spread based on halfspaces and hyperplanes of the Newton polytope, and we classify basic monomial ideals. We then apply this method to calculate the analytic spread for a few families of monomial ideals.

Paper Structure

This paper contains 10 sections, 28 theorems, 24 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Bounding the analytic spread with the Newton polytope
  4. Basic monomial ideals
  5. On the Equality of Theorem \ref{['thm:spreadCalc']}
  6. Three-dimensional Case
  7. Ideals with Disjointly Generated Primary Decomposition
  8. Intersections of powers of two monomial primes
  9. Open Questions
  10. Monomial Reductions

Key Result

Theorem 1

Let $I$ be a monomial ideal, and let $\operatorname{np}(I)$ have hyperplanes defined by and halfspaces defined by Suppose that there exist $\alpha_1,\dots,\alpha_s\in\mathbb{R}$ and $\beta_1,\dots,\beta_t\in\mathbb{R}_{\geq 0}$ such that has all negative entries. Then $\ell(I)\leq n+1-(s+k)$ where $k$ is the minimal number of non-zero $\beta_j$ required to achieve $\mathbf{W}$ with all negative

Figures (3)

  • Figure 1: ${\rm NP}(xy, yz, xz)$
  • Figure 2: $\operatorname{NP}(xy, y^4z^4, x^4z^4)$
  • Figure 3: Newton polytope and polyhedron of $I$ from Remark \ref{['rem:nonEqual']}

Theorems & Definitions (60)

  • Theorem 1: Theorem \ref{['thm:spreadCalc']}
  • Theorem 2: Theorem \ref{['thm:expectedCodim']}
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: Corollary 8.3.6, Proposition 8.3.7 HS
  • Corollary 2.6
  • Definition 2.7
  • Theorem 2.8: Bivià-Ausina BA
  • ...and 50 more