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Waldschmidt constant of monomial ideals and Simis ideals

Bijender, Ajay Kumar

TL;DR

The work investigates lower bounds for the Waldschmidt constant of monomial ideals and the Simis conjecture in a combinatorial setting. By leveraging standard linear weightings and polarization, it reduces questions to squarefree cases and preserves key invariants such as $\rho$ and $\rho_a$, enabling a proof of $\hat{\alpha}(I) \ge \dfrac{\alpha(I) + h - 1}{h}$ for broad classes (Theorem Th-Waldschmidt). It further verifies the Mendez–Pinto–Villarreal conjecture $C_2$ for several classes of monomial ideals via explicit combinatorial criteria, including a height-2 condition and a general whisker-based configuration, establishing equivalences with Simis-ness under certain weightings. The results extend known squarefree cases, providing concrete criteria for identifying Simis ideals and connecting symbolic-power behavior to hypergraph structure.

Abstract

In 2017, Cooper et al. proposed a conjecture providing a lower bound for the Waldschmidt constant of monomial ideals. We confirm this conjecture for some classes of monomial ideals. Recently, Méndez, Pinto, and Villarreal formulated a conjecture stating that if $I$ is a monomial ideal without embedded associated primes, whose irreducible decomposition is minimal and which is a Simis ideal, then there exist a Simis squarefree monomial ideal $J$ and a standard linear weighting $w$ such that $I = J_{w}.$ In this work, we verify this conjecture for some classes of monomial ideals.

Waldschmidt constant of monomial ideals and Simis ideals

TL;DR

The work investigates lower bounds for the Waldschmidt constant of monomial ideals and the Simis conjecture in a combinatorial setting. By leveraging standard linear weightings and polarization, it reduces questions to squarefree cases and preserves key invariants such as and , enabling a proof of for broad classes (Theorem Th-Waldschmidt). It further verifies the Mendez–Pinto–Villarreal conjecture for several classes of monomial ideals via explicit combinatorial criteria, including a height-2 condition and a general whisker-based configuration, establishing equivalences with Simis-ness under certain weightings. The results extend known squarefree cases, providing concrete criteria for identifying Simis ideals and connecting symbolic-power behavior to hypergraph structure.

Abstract

In 2017, Cooper et al. proposed a conjecture providing a lower bound for the Waldschmidt constant of monomial ideals. We confirm this conjecture for some classes of monomial ideals. Recently, Méndez, Pinto, and Villarreal formulated a conjecture stating that if is a monomial ideal without embedded associated primes, whose irreducible decomposition is minimal and which is a Simis ideal, then there exist a Simis squarefree monomial ideal and a standard linear weighting such that In this work, we verify this conjecture for some classes of monomial ideals.
Paper Structure (4 sections, 12 theorems, 71 equations)

This paper contains 4 sections, 12 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Waldschmidt Constant of a Standard Weighted Monomial Ideal
  4. The Mendez-Pinto-Villarreal Conjecture

Key Result

Theorem 3.3

Let $I \subset R$ be a monomial ideal, and let $w$ be a standard linear weighting on $R.$ Then $\rho(I_w) = \rho(I)$ and $\rho_a(I_w) = \rho_a(I).$ Moreover, if $h$ is the big-height of $I$ and $I$ has a standard linear weighting, then

Theorems & Definitions (41)

  • Conjecture 1.1
  • Conjecture 1.2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 31 more