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Polarization and spreading of monomial ideals

Mircea Cimpoeas

Abstract

We characterize the monomial ideals $I\subset K[x_1,\ldots,x_n]$ with the property that the polarization $I^p$ and $I^{σ^n}:=$ the ideal obtained from $I$ by the $n$-th iterated squarefree operator $σ$ are isomorphic via a permutation of variables. We give several methods to construct such ideals. We also compare the depth and sdepth of $I$ and $I^{σ^n}$.

Polarization and spreading of monomial ideals

Abstract

We characterize the monomial ideals with the property that the polarization and the ideal obtained from by the -th iterated squarefree operator are isomorphic via a permutation of variables. We give several methods to construct such ideals. We also compare the depth and sdepth of and .

Paper Structure

This paper contains 2 sections, 17 theorems, 103 equations.

Table of Contents

  1. Main results
  2. Lcm-lattices and spreading of monomial ideals

Key Result

Lemma 1.1

Let $u,v\in T_n$ be two monomials and let $t\geq n$ be an integer. If $\operatorname{gcd}(u,v)=1$ then $\operatorname{gcd}(\sigma^t(u),\sigma^t(v))=1$.

Theorems & Definitions (42)

  • Lemma 1.1
  • proof
  • Proposition 1.2
  • proof
  • Remark 1.3
  • Proposition 1.4
  • proof
  • Corollary 1.5
  • proof
  • Remark 1.6
  • ...and 32 more