{
  "abstract": "In this paper, we introduce several novel approaches utilizing the circle method to obtain the asymptotic formula for the number of integral points of bounded height lying on a hypersurface in a weighted projective space. Let $F(\\mathbf{x} ; \\mathbf{y})$ be a given weighted form of degree $d$ in variables $\\mathbf{x} \\in \\mathbb{R}^{s_1}$ and $\\mathbf{y} \\in \\mathbb{R}^{s_2}$, where variables $\\mathbf{x}$ and $\\mathbf{y}$ have weights $w_1$ and $w_2$ with $w_1<w_2,\\left(w_1, w_2\\right)=1$, and $d>w_1 w_2$. Write $$ R_F(P):=\\#\\left\\{(\\mathbf{x} ; \\mathbf{y}) \\in \\mathbb{Z}^{s_1+s_2}: F(\\mathbf{x} ; \\mathbf{y})=0,|\\mathbf{x}| \\leq P^{w_1 / d},|\\mathbf{y}| \\leq P^{w_2 / d}\\right\\} . $$ In particular, we show that whenever $$ s_1+s_2-σ_F>\\left(1+\\frac{w_2}{w_1}\\right) \\frac{d}{w_1} 2^{d / w_1}, $$ where $σ_F$ is the dimension of the affine singular locus of $F$, the quantity $R_F(P)$ has the expected asymptotic formula, that is $$ R_F(P)=c P^{s_1 w_1 / d+s_2 w_2 / d-1}+o\\left(P^{s_1 w_1 / d+s_2 w_2 / d-1}\\right), $$ where $c$ is the product of local densities. Furthermore, the constant $c$ is positive whenever $F(\\mathbf{x} ; \\mathbf{y})=0$ has a nonsingular solution over $\\mathbb{R}$ and $\\mathbb{Q}_p$ for every prime $p$. As a corollary, we verify the integral Hasse principle for the quasi-smooth hypersurface defined by $F(\\mathbf{x} ; \\mathbf{y})=0$ in a weighted projective space of sufficiently large dimensions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28774",
  "author_total": 2,
  "authors": [
    {
      "name": "Daniel Flores Galiote"
    },
    {
      "name": "Kiseok Yeon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2gav715zynh6n.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28774",
  "fulltext_url": null,
  "id": "2gav715zynh6n",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.28774",
  "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.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7204936176",
  "pdf_url": "https://arxiv.org/pdf/2608.28774",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2gav715zynh6n.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.28774",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28774"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On weighted forms in many variables",
  "title_text": "On weighted forms in many variables",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/2gav715zynh6n",
  "venue": "arXiv",
  "versions": []
}