{
  "abstract": "Suppose $f_1, \\dots, f_{n+1}$ are elements of a regular ring $R$, where $n := \\dim R$. The first author proved that $f_1^n \\cdots f_{n+1}^n \\in (f_1^{n+1}, \\dots, f_{n+1}^{n+1})R$; this uses the Brian\\c con-Skoda theorem --- no simpler proof is known, as far as we are aware. We show here that this result is optimal in many respects. On the other hand, when the $f_i$ are homogeneous general polynomials of a fixed degree in $R := \\mathbb{F}[x_1,\\dots,x_n]$, for $\\mathbb{F}$ a field of characteristic zero, we prove that $f_1 \\cdots f_{n+1} \\in (f_1^2, \\dots, f_{n+1}^2)R$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10200",
  "author_total": 4,
  "authors": [
    {
      "name": "Melvin Hochster"
    },
    {
      "name": "Jack Jeffries"
    },
    {
      "name": "Nawaj KC"
    },
    {
      "name": "Anurag K. Singh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bh8qwyhakm8aj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10200",
  "fulltext_url": null,
  "id": "bh8qwyhakm8aj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10200",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7220923464",
  "pdf_url": "https://arxiv.org/pdf/2610.10200",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bh8qwyhakm8aj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.10200",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10200"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On an ideal membership problem",
  "title_text": "On an ideal membership problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/bh8qwyhakm8aj",
  "venue": "arXiv",
  "versions": []
}