---
_id: '256'
abstract:
- lang: eng
  text: We show that a non-singular integral form of degree d is soluble over the
    integers if and only if it is soluble over ℝ and over ℚp for all primes p, provided
    that the form has at least (d - 1/2 √d)2d variables. This improves on a longstanding
    result of Birch.
acknowledgement: While working on this paper the authors were supported by the Leverhulme
  Trust and ERC grant 306457.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Timothy D
  full_name: Browning, Timothy D
  id: 35827D50-F248-11E8-B48F-1D18A9856A87
  last_name: Browning
  orcid: 0000-0002-8314-0177
- first_name: Sean
  full_name: Prendiville, Sean
  last_name: Prendiville
citation:
  ama: Browning TD, Prendiville S. Improvements in Birch’s theorem on forms in many
    variables. <i>Journal fur die Reine und Angewandte Mathematik</i>. 2017;2017(731):122.
    doi:<a href="https://doi.org/10.1515/crelle-2014-0122">10.1515/crelle-2014-0122</a>
  apa: Browning, T. D., &#38; Prendiville, S. (2017). Improvements in Birch’s theorem
    on forms in many variables. <i>Journal Fur Die Reine Und Angewandte Mathematik</i>.
    Walter de Gruyter. <a href="https://doi.org/10.1515/crelle-2014-0122">https://doi.org/10.1515/crelle-2014-0122</a>
  chicago: Browning, Timothy D, and Sean Prendiville. “Improvements in Birch’s Theorem
    on Forms in Many Variables.” <i>Journal Fur Die Reine Und Angewandte Mathematik</i>.
    Walter de Gruyter, 2017. <a href="https://doi.org/10.1515/crelle-2014-0122">https://doi.org/10.1515/crelle-2014-0122</a>.
  ieee: T. D. Browning and S. Prendiville, “Improvements in Birch’s theorem on forms
    in many variables,” <i>Journal fur die Reine und Angewandte Mathematik</i>, vol.
    2017, no. 731. Walter de Gruyter, p. 122, 2017.
  ista: Browning TD, Prendiville S. 2017. Improvements in Birch’s theorem on forms
    in many variables. Journal fur die Reine und Angewandte Mathematik. 2017(731),
    122.
  mla: Browning, Timothy D., and Sean Prendiville. “Improvements in Birch’s Theorem
    on Forms in Many Variables.” <i>Journal Fur Die Reine Und Angewandte Mathematik</i>,
    vol. 2017, no. 731, Walter de Gruyter, 2017, p. 122, doi:<a href="https://doi.org/10.1515/crelle-2014-0122">10.1515/crelle-2014-0122</a>.
  short: T.D. Browning, S. Prendiville, Journal Fur Die Reine Und Angewandte Mathematik
    2017 (2017) 122.
date_created: 2018-12-11T11:45:28Z
date_published: 2017-10-01T00:00:00Z
date_updated: 2024-03-05T12:09:21Z
day: '01'
doi: 10.1515/crelle-2014-0122
extern: '1'
external_id:
  arxiv:
  - '1402.4489'
intvolume: '      2017'
issue: '731'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1402.4489
month: '10'
oa: 1
oa_version: Preprint
page: '122'
publication: Journal fur die Reine und Angewandte Mathematik
publication_identifier:
  issn:
  - 0075-4102
publication_status: published
publisher: Walter de Gruyter
publist_id: '7646'
quality_controlled: '1'
related_material:
  record:
  - id: '271'
    relation: earlier_version
    status: public
status: public
title: Improvements in Birch's theorem on forms in many variables
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 2017
year: '2017'
...
---
_id: '271'
abstract:
- lang: eng
  text: We show that a non-singular integral form of degree d is soluble non-trivially
    over the integers if and only if it is soluble non-trivially over the reals and
    the p-adic numbers, provided that the form has at least (d-\sqrt{d}/2)2^d variables.
    This improves on a longstanding result of Birch.
acknowledgement: While working on this paper the authors were supported by the Leverhulme
  Trust and ERC grant 306457.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Timothy D
  full_name: Browning, Timothy D
  id: 35827D50-F248-11E8-B48F-1D18A9856A87
  last_name: Browning
  orcid: 0000-0002-8314-0177
- first_name: Sean
  full_name: Prendiville, Sean
  last_name: Prendiville
citation:
  ama: Browning TD, Prendiville S. Improvements in Birch’s theorem on forms in many
    variables. <i>Journal fur die Reine und Angewandte Mathematik</i>. 2017(731):203-234.
    doi:<a href="https://doi.org/10.1515/crelle-2014-0122">10.1515/crelle-2014-0122</a>
  apa: Browning, T. D., &#38; Prendiville, S. (n.d.). Improvements in Birch’s theorem
    on forms in many variables. <i>Journal Fur Die Reine Und Angewandte Mathematik</i>.
    Walter de Gruyter. <a href="https://doi.org/10.1515/crelle-2014-0122">https://doi.org/10.1515/crelle-2014-0122</a>
  chicago: Browning, Timothy D, and Sean Prendiville. “Improvements in Birch’s Theorem
    on Forms in Many Variables.” <i>Journal Fur Die Reine Und Angewandte Mathematik</i>.
    Walter de Gruyter, n.d. <a href="https://doi.org/10.1515/crelle-2014-0122">https://doi.org/10.1515/crelle-2014-0122</a>.
  ieee: T. D. Browning and S. Prendiville, “Improvements in Birch’s theorem on forms
    in many variables,” <i>Journal fur die Reine und Angewandte Mathematik</i>, vol.
    2017, no. 731. Walter de Gruyter, pp. 203–234.
  ista: Browning TD, Prendiville S. Improvements in Birch’s theorem on forms in many
    variables. Journal fur die Reine und Angewandte Mathematik. 2017(731), 203–234.
  mla: Browning, Timothy D., and Sean Prendiville. “Improvements in Birch’s Theorem
    on Forms in Many Variables.” <i>Journal Fur Die Reine Und Angewandte Mathematik</i>,
    vol. 2017, no. 731, Walter de Gruyter, pp. 203–34, doi:<a href="https://doi.org/10.1515/crelle-2014-0122">10.1515/crelle-2014-0122</a>.
  short: T.D. Browning, S. Prendiville, Journal Fur Die Reine Und Angewandte Mathematik
    2017 (n.d.) 203–234.
date_created: 2018-12-11T11:45:32Z
date_published: 2015-02-20T00:00:00Z
date_updated: 2024-03-05T12:09:22Z
day: '20'
doi: 10.1515/crelle-2014-0122
extern: '1'
external_id:
  arxiv:
  - '1402.4489'
intvolume: '      2017'
issue: '731'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1402.4489
month: '02'
oa: 1
oa_version: Preprint
page: 203 - 234
publication: Journal fur die Reine und Angewandte Mathematik
publication_identifier:
  issn:
  - 0075-4102
publication_status: submitted
publisher: Walter de Gruyter
publist_id: '7631'
quality_controlled: '1'
related_material:
  record:
  - id: '256'
    relation: later_version
    status: public
status: public
title: Improvements in Birch's theorem on forms in many variables
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 2017
year: '2015'
...