{
  "abstract": "Given a system $\\mathbf{f}=(f_1,\\ldots,f_n)$ of $n$ homogeneous forms in $n$ variables of the same degree, Macaulay's resultant vanishes precisely when the polynomials have a common projective zero. Its order of vanishing measures the singularity of the resultant hypersurface at $\\mathbf{f}$. In this paper, we study how this multiplicity reflects the geometry of the projective zero scheme defined by $\\mathbf{f}$. We give an exact formula for the multiplicity, expressed in terms of the projective degrees of the rational map defined by $\\mathbf{f}$. As a consequence, we obtain a geometric lower bound involving the degrees, dimensions, and multiplicities of the irreducible components of the projective zero scheme. This extends the multiplicity estimates of Roy and Ghidelli from zero-dimensional schemes to schemes of arbitrary dimension. Finally, we apply this geometric estimate to tensor eigenvalues. It translates directly into a lower bound for the algebraic multiplicity of a tensor eigenvalue in terms of the geometry of its eigenscheme. This settles a conjecture by Canino et al. and consequently settles earlier conjectures of Qi and of Hu and Ye concerning the relationship between algebraic, geometric, and span multiplicities of tensor eigenvalues.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01268",
  "author_total": 3,
  "authors": [
    {
      "name": "Mahmut Levent Doğan"
    },
    {
      "name": "Elias Tsigaridas"
    },
    {
      "name": "Zafeirakis Zafeirakopoulos"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8fysm1e1yhf0b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01268",
  "fulltext_url": null,
  "id": "8fysm1e1yhf0b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01268",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2605|Tensor decomposition and applications"
    ]
  ],
  "openalex": "W7206195368",
  "pdf_url": "https://arxiv.org/pdf/2609.01268",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8fysm1e1yhf0b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.01268",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01268"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Resultant multiplicity via projective degrees and applications to tensor eigenvalues",
  "title_text": "Resultant multiplicity via projective degrees and applications to tensor eigenvalues",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/8fysm1e1yhf0b",
  "venue": "arXiv",
  "versions": []
}