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arXiv · 2505.21802

The Minimum Number of Generators of Symmetric Ideals

Noah Walker
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Abstract

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.

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Noah Walker. 2025-05-27. The Minimum Number of Generators of Symmetric Ideals. https://arxiv.org/abs/2505.21802

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