Noah Walker | Searcharxiv

arXiv2025

The Minimum Number of Generators of Symmetric Ideals

We study equigenerated symmetric ideals in polynomial rings and the minimum number of polynomials required to generate them up to permutations of the variables. We give a representation-theoretic formula for this number and determine a sharp threshold on the number of variables needed for a general symmetric ideal to attain the largest possible dimension in its generating degree. The threshold is governed by partial sums of integer partition numbers, which appear in the OEIS as sequence A000070. We also construct explicit extremal symmetric ideals for every admissible number of generators. As an application, the principal case gives the sharp stable range for the theorem of Harada-Seceleanu-\c{S}ega on general principal symmetric ideals.

math.AC↗

arXiv2024

Products and powers of principal symmetric ideals

Principal symmetric ideals were recently introduced by Harada, Seceleanu, and Sega, with a focus on their homological properties. They are ideals generated by the orbit of a single polynomial under permutations of variables in a polynomial ring. In this paper we seek to determine when a product of two principal symmetric ideals is principal symmetric and when all the powers of a principal symmetric ideal are again principal symmetric ideals. We characterize the ideals that have the latter property as being generated by polynomials invariant up to a scalar multiple under permutation of variables. Recognizing principal symmetric ideals is an open question for the purpose of which we produce certain obstructions. We also demonstrate that the Hilbert functions of symmetric monomial ideals are not all given by symmetric monomial ideals, in contrast to the non-symmetric case.

math.AC↗