On General Principal Symmetric Ideals
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917416471101440 |
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| author | Walker, Noah |
| author_facet | Walker, Noah |
| contents | In a recent paper by Harada, Seceleanu, and Şega, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_21802 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On General Principal Symmetric Ideals Walker, Noah Commutative Algebra 13A50, 20C30 (Primary) 05E40, 13C05 (Secondary) In a recent paper by Harada, Seceleanu, and Şega, 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. |
| title | On General Principal Symmetric Ideals |
| topic | Commutative Algebra 13A50, 20C30 (Primary) 05E40, 13C05 (Secondary) |
| url | https://arxiv.org/abs/2505.21802 |