---
OA_place: repository
OA_type: green
_id: '21768'
abstract:
- lang: eng
  text: Let F∈Z[x1,…,xn] be a homogeneous form of degree d≥2, and V∗F the singular
    locus of the hypersurface {x∈AnC:F(x)=0}. A longstanding result of Birch states
    that there is a non-trivial integral solution to the equation F(x1,…,xn)=0 provided
    n>dimV∗F+(d−1)2d, and there is a non-singular solution in R and Qp for all primes
    p. We give a different formulation of this result. More precisely, we replace
    dimV∗F with a quantity HF defined in terms of the Hessian matrix of F. This quantity
    satisfies 0≤HF≤dimV∗F; therefore, we improve on the aforementioned result of Birch
    if HF<dimV∗F. We also prove the corresponding result for systems of forms of equal
    degree.
acknowledgement: The author would like to thank Tim Browning, Jakob Glas and Simon
  Rydin Myerson for useful suggestions and conversations. Finally, he would like to
  thank the anonymous referees for their helpful comments. The author was supported
  by the NWO Veni Grant 016.Veni.192.047 during his time at Utrecht University and
  by the FWF grant P 36278 at the Institute of Science and Technology Austria while
  working on this article.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Shuntaro
  full_name: Yamagishi, Shuntaro
  id: 0c3fbc5c-f7a6-11ec-8d70-9485e75b416b
  last_name: Yamagishi
citation:
  ama: Yamagishi S. Birch’s theorem on forms in many variables with a Hessian condition.
    <i>Acta Arithmetica</i>. 2025;221(2):141-151. doi:<a href="https://doi.org/10.4064/aa241029-19-8">10.4064/aa241029-19-8</a>
  apa: Yamagishi, S. (2025). Birch’s theorem on forms in many variables with a Hessian
    condition. <i>Acta Arithmetica</i>. Instytut Matematyczny. <a href="https://doi.org/10.4064/aa241029-19-8">https://doi.org/10.4064/aa241029-19-8</a>
  chicago: Yamagishi, Shuntaro. “Birch’s Theorem on Forms in Many Variables with a
    Hessian Condition.” <i>Acta Arithmetica</i>. Instytut Matematyczny, 2025. <a href="https://doi.org/10.4064/aa241029-19-8">https://doi.org/10.4064/aa241029-19-8</a>.
  ieee: S. Yamagishi, “Birch’s theorem on forms in many variables with a Hessian condition,”
    <i>Acta Arithmetica</i>, vol. 221, no. 2. Instytut Matematyczny, pp. 141–151,
    2025.
  ista: Yamagishi S. 2025. Birch’s theorem on forms in many variables with a Hessian
    condition. Acta Arithmetica. 221(2), 141–151.
  mla: Yamagishi, Shuntaro. “Birch’s Theorem on Forms in Many Variables with a Hessian
    Condition.” <i>Acta Arithmetica</i>, vol. 221, no. 2, Instytut Matematyczny, 2025,
    pp. 141–51, doi:<a href="https://doi.org/10.4064/aa241029-19-8">10.4064/aa241029-19-8</a>.
  short: S. Yamagishi, Acta Arithmetica 221 (2025) 141–151.
corr_author: '1'
das_tickbox: '0'
date_created: 2026-04-26T22:01:48Z
date_published: 2025-10-28T00:00:00Z
date_updated: 2026-07-16T09:04:43Z
day: '28'
department:
- _id: TiBr
doi: 10.4064/aa241029-19-8
external_id:
  arxiv:
  - '2304.02620'
intvolume: '       221'
issue: '2'
keyword:
- Diophantine equations
- homogeneous forms
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2304.02620
month: '10'
oa: 1
oa_version: Preprint
page: 141-151
project:
- _id: bd8a4fdc-d553-11ed-ba76-80a0167441a3
  grant_number: P36278
  name: Rational curves via function field analytic number theory
publication: Acta Arithmetica
publication_identifier:
  eissn:
  - 1730-6264
  issn:
  - 0065-1036
publication_status: published
publisher: Instytut Matematyczny
quality_controlled: '1'
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: no
title: Birch’s theorem on forms in many variables with a Hessian condition
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 221
year: '2025'
...