Asymptotic Degree of Random Monomial Ideals

TL;DR

This paper investigates the asymptotic behavior of the degree of random monomial ideals, revealing a hyperbolic staircase shape and connecting it to number theory, with bounds on standard pairs providing detailed invariant analysis.

Contribution

It introduces a probabilistic analysis of the degree of random monomial ideals, characterizes the staircase shape as hyperbolic, and establishes bounds on standard pairs for these ideals.

Findings

The staircase diagram of a random monomial ideal is approximately hyperbolic.

The asymptotic degree relates to the discrete volume under the staircase, connected to divisor functions.

Bounds are derived for the number of standard pairs, enriching the understanding of the ideal's invariants.

Abstract

One of the fundamental invariants connecting algebra and geometry is the degree of an ideal. In this paper we derive the probabilistic behavior of degree with respect to the versatile Erd\H{o}s-R\'enyi-type model for random monomial ideals defined in \cite{rmi}. We study the staircase structure associated to a monomial ideal, and show that in the random case the shape of the staircase diagram is approximately hyperbolic, and this behavior is robust across several random models. Since the discrete volume under this staircase is related to the summatory higher-order divisor function studied in number theory, we use this connection and our control over the shape of the staircase diagram to derive the asymptotic degree of a random monomial ideal. Another way to compute the degree of a monomial ideal is with a standard pair decomposition. This paper derives bounds on the number of standard pairs of a random monomial ideal indexed by any subset of the ring variables. The standard pairs indexed by maximal subsets give a count of degree, as well as being a more nuanced invariant of the random monomial ideal.