{
  "abstract": "The external zonotopal algebra of a graph is generated by nilpotent vertex elements $X_v$ and determines the graphical matroid after deleting loops. Replacing these generators by $f(X_v)$, for a power series $f(u)$, and filtering by word length gives a family of filtered algebras. We extend filtered graph reconstruction from simple graphs to loopless multigraphs without isolated vertices when the linear and quadratic coefficients of $f$ are nonzero. Nevertheless, even the family of Hilbert functions indexed by all polynomials does not determine the graph: two nonisomorphic ten-vertex trees have the same Hilbert function for every $f$, answering a question of Nenashev and the first author. For forests and series whose first nonconstant term has degree $k\\ge 2$, we give a monomial presentation indexed by partial score vectors whose entries are divisible by $k$. For normalized nondegenerate functions, a boundary-state calculus gives a common finite parameter partition for all trees of a fixed maximum degree. In maximum degrees three and four, exact symbolic calculations identify these partitions by the vanishing of specified coefficients of $f^{-1}$. For functions with nonzero quadratic coefficient, the same calculus gives explicit recurrences for attached or connecting paths. Symmetry reductions, including commuting actions of $sl_2$ and the symmetric group on the leaves of a star, determine the generic Hilbert series of stars and two-vertex multigraphs and also apply to parallel edges and leaves in larger graphs. For cycles, we determine the highest-degree term and the first three nonconstant coefficients in the stable range. A weighted independent-set formula describes the trace sequence in the remaining conjecture.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38469",
  "author_total": 3,
  "authors": [
    {
      "name": "Boris Shapiro"
    },
    {
      "name": "Ilya Smirnov"
    },
    {
      "name": "Arkady Vaintrob"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d8rxsw5m4dwks.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38469",
  "fulltext_url": null,
  "id": "d8rxsw5m4dwks",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38469",
  "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"
    ],
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38469",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d8rxsw5m4dwks.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38469",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38469"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Hilbert functions of filtered graphical zonotopal algebras",
  "title_text": "Hilbert functions of filtered graphical zonotopal algebras",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/d8rxsw5m4dwks",
  "venue": "arXiv",
  "versions": []
}