On General Principal Symmetric Ideals

TL;DR

This paper improves bounds on the number of variables needed to determine properties of general principal symmetric ideals, extending prior results with explicit bounds and new structural insights.

Contribution

It provides an effective bound on variables for principal symmetric ideals and introduces maximal r-generated submodules with their structural characterization.

Findings

Established a bound related to partition numbers for principal symmetric ideals.

Proved a recognition theorem for principal symmetric ideals.

Connected maximal r-generated submodules to general symmetric ideals.

Abstract

In a recent paper by Harada, Seceleanu, and \c{S}ega, the Hilbert function, betti table, and graded minimal free resolution of a general principal symmetric ideal are determined when the number of variables in the polynomial ring is sufficiently large. In this paper, we strengthen that result by giving a effective bound on the number of variables needed for their conclusion to hold. The bound is related to a well-known integer sequence involving partition numbers (OEIS A000070). Along the way, we prove a recognition theorem for principal symmetric ideals. We also introduce the class of maximal $r$-generated submodules, determine their structure, and connect them to general symmetric ideals.