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The Weak Lefschetz property and unimodality of Hilbert functions of random monomial algebras

Uwe Nagel, Sonja Petrović

Abstract

In this work, we investigate the presence of the weak Lefschetz property (WLP) and Hilbert functions for various types of random standard graded Artinian algebras. If an algebra has the WLP then its Hilbert function is unimodal. Using probabilistic models for random monomial algebras, our results and simulations suggest that in each considered regime the Hilbert functions of the produced algebras are unimodal with high probability. The WLP appears to be present with high probability most of the time. However, we propose that there is one scenario where the generated algebras fail to have the WLP with high probability.

The Weak Lefschetz property and unimodality of Hilbert functions of random monomial algebras

Abstract

In this work, we investigate the presence of the weak Lefschetz property (WLP) and Hilbert functions for various types of random standard graded Artinian algebras. If an algebra has the WLP then its Hilbert function is unimodal. Using probabilistic models for random monomial algebras, our results and simulations suggest that in each considered regime the Hilbert functions of the produced algebras are unimodal with high probability. The WLP appears to be present with high probability most of the time. However, we propose that there is one scenario where the generated algebras fail to have the WLP with high probability.
Paper Structure (9 sections, 10 theorems, 59 equations, 6 figures, 7 tables)

This paper contains 9 sections, 10 theorems, 59 equations, 6 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Background and notation: level algebras, weak Lefschetz property, randomness, and sampling
  3. Random standard graded Artinian algebras
  4. Simulation results
  5. Expected Hilbert functions
  6. WLP simulations
  7. Pure $O$-sequences
  8. WLP for Random level algebras
  9. Simulation results

Key Result

Theorem 2.2

Let $A$ be a standard graded Artinian algebra. Let $L\in A_1$ be a general linear form and for $d\ge0$ consider the map $\phi_d:A_{d}\to A_{d+1}$ defined by multiplication by $L$.

Figures (6)

  • Figure 1: Expected Hilbert functions for ${I}\sim\mathcal{I}(n,D,p)$. Left: $D$ values of $p$ ranging from $1/D$ to $1$. Right: 10 selected values of $p$, with legend. Note that each curve, of a fixed color, represents one sample of size $N=100$ for one value of $p$.
  • Figure 2: Expected Hilbert functions $h_d(S/{I})$ of monomial ideals ${I}\sim\mathcal{I}(n,D,p)$. Each color curve represents one sample of size $N=100$ for one value of $p$. The top left figure shows the legend for values of the probability parameter $p$. The remaining four figures have about 20 curves so the legends are hidden.
  • Figure 3: Expected Hilbert functions Artinian algebras ${I}+(x_1^D,\dots,x_n^D)$, where ${I} \sim \mathcal{I}(n,D,p)$. Since the figure on the left gives a global trend, the middle and right figure offer a zoomed-in view of the sample (note the truncated $y$-axes in the middle and right figures).
  • Figure 4: Expected Hilbert functions for select random Artinian algebras ${I}+(x_1^D,\dots,x_n^D)$, where ${I} \sim \mathcal{I}(n,D,p)$, obtained from the same samples of monomial algebras from Table \ref{['table:RandomArtinUnimodalStats']}. The figure on the right represents 50 algebras generated using the values $p=0.01$ and $p=0.02$, while the figure on the left values $p>0.1$.
  • Figure 5: Expected Hilbert functions for select random Artinian algebras ${I}+(x_1^D,\dots,x_n^D)$, where ${I} \sim \mathcal{I}(n,D,p)$. The figure on the left represents algebras generated using the value of $p$ approximately $1/D$, while on the right $1/D^2$ and $1/D^3$. The samples are selected from the same samples of monomial algebras from Table \ref{['table:RandomArtinUnimodalStats']}.
  • ...and 1 more figures

Theorems & Definitions (27)

  • Definition 2.1
  • Theorem 2.2: MMN
  • Proposition 2.3: MMN
  • Example 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Proposition 3.4
  • proof
  • Corollary 3.5
  • ...and 17 more