---
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'
fulldoi: https://doi.org/10.4064/aa241029-19-8
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'
...
---
OA_type: closed access
_id: '20603'
abstract:
- lang: eng
  text: "We study the growth of sumsets A+B⊂S⊂G, where S does not contain an arithmetic
    progression of length 2k+1, and where G is a commutative group, in which every
    nonzero element has an order of at least 2k+1. More specifically, we show the
    following: if A,B⊂G are sets such that A+B does not contain an arithmetic progression
    of length 2k+1, then\r\n|A+B|≥|A|2k−13k−2|B|k3k−2.\r\nAs an application we derive
    upper bounds on the cardinality of the summands in sumsets A+B+C contained in
    the set of t-th powers, where t≥2 is an integer. In particular, we show that min(|A|,|B|,|C|)≪(logN)4/5
    for t=2, and min(|A|,|B|,|C|)≪t(logN)1/2 for t≥3."
acknowledgement: "The authors would like to thank the referee and Ilya Shkredov for
  comments on the manuscript.\r\nC. E. is supported by a joint FWF-ANR project ArithRand,
  grant numbers FWF I 4945-N and ANR-20-CE91-0006.\r\n"
article_processing_charge: No
article_type: original
author:
- first_name: Christian
  full_name: Elsholtz, Christian
  last_name: Elsholtz
- first_name: Imre Z.
  full_name: Ruzsa, Imre Z.
  last_name: Ruzsa
- first_name: Lena
  full_name: Wurzinger, Lena
  id: 50c57d72-32a8-11ee-aeea-d652094d2ccd
  last_name: Wurzinger
  orcid: 0009-0004-5360-0074
citation:
  ama: Elsholtz C, Ruzsa IZ, Wurzinger L. Sumset growth in progression-free sets.
    <i>Acta Arithmetica</i>. 2025;220:289-303. doi:<a href="https://doi.org/10.4064/aa250115-14-7">10.4064/aa250115-14-7</a>
  apa: Elsholtz, C., Ruzsa, I. Z., &#38; Wurzinger, L. (2025). Sumset growth in progression-free
    sets. <i>Acta Arithmetica</i>. Institute of Mathematics. <a href="https://doi.org/10.4064/aa250115-14-7">https://doi.org/10.4064/aa250115-14-7</a>
  chicago: Elsholtz, Christian, Imre Z. Ruzsa, and Lena Wurzinger. “Sumset Growth
    in Progression-Free Sets.” <i>Acta Arithmetica</i>. Institute of Mathematics,
    2025. <a href="https://doi.org/10.4064/aa250115-14-7">https://doi.org/10.4064/aa250115-14-7</a>.
  ieee: C. Elsholtz, I. Z. Ruzsa, and L. Wurzinger, “Sumset growth in progression-free
    sets,” <i>Acta Arithmetica</i>, vol. 220. Institute of Mathematics, pp. 289–303,
    2025.
  ista: Elsholtz C, Ruzsa IZ, Wurzinger L. 2025. Sumset growth in progression-free
    sets. Acta Arithmetica. 220, 289–303.
  mla: Elsholtz, Christian, et al. “Sumset Growth in Progression-Free Sets.” <i>Acta
    Arithmetica</i>, vol. 220, Institute of Mathematics, 2025, pp. 289–303, doi:<a
    href="https://doi.org/10.4064/aa250115-14-7">10.4064/aa250115-14-7</a>.
  short: C. Elsholtz, I.Z. Ruzsa, L. Wurzinger, Acta Arithmetica 220 (2025) 289–303.
corr_author: '1'
das_tickbox: '0'
date_created: 2025-11-04T14:33:16Z
date_published: 2025-09-12T00:00:00Z
date_updated: 2026-07-29T09:51:25Z
day: '12'
department:
- _id: TiBr
doi: 10.4064/aa250115-14-7
external_id:
  isi:
  - '001570716800001'
fulldoi: https://doi.org/10.4064/aa250115-14-7
intvolume: '       220'
isi: 1
language:
- iso: eng
month: '09'
oa_version: None
page: 289-303
publication: Acta Arithmetica
publication_identifier:
  eissn:
  - 1730-6264
  issn:
  - 0065-1036
publication_status: published
publisher: Institute of Mathematics
quality_controlled: '1'
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: no
title: Sumset growth in progression-free sets
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 220
year: '2025'
...
---
_id: '12406'
abstract:
- lang: eng
  text: Let X be a sufficiently large positive integer. We prove that one may choose
    a subset S of primes with cardinality O(logX) such that a positive proportion
    of integers less than X can be represented by x2+py2 for at least one p∈S.
acknowledgement: "This article is a version the author’s master thesis at the University
  of Bonn. The author would like to thank his advisor Valentin Blomer for introducing
  the problem, and giving generous feedback and encouragement along the way, especially
  during the global pandemic.\r\nThe author thanks Edgar Assing for his lectures on
  analytic number theory. Finally, the author is grateful to the anonymous referees
  for their valuable time and comments.\r\n"
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Yijie
  full_name: Diao, Yijie
  id: 7b7eb4ca-eb2c-11ec-b98b-accec0b20c3b
  last_name: Diao
  orcid: 0000-0002-4989-5330
citation:
  ama: Diao Y. Density of the union of positive diagonal binary quadratic forms. <i>Acta
    Arithmetica</i>. 2023;207:1-17. doi:<a href="https://doi.org/10.4064/aa210830-24-11">10.4064/aa210830-24-11</a>
  apa: Diao, Y. (2023). Density of the union of positive diagonal binary quadratic
    forms. <i>Acta Arithmetica</i>. Instytut Matematyczny. <a href="https://doi.org/10.4064/aa210830-24-11">https://doi.org/10.4064/aa210830-24-11</a>
  chicago: Diao, Yijie. “Density of the Union of Positive Diagonal Binary Quadratic
    Forms.” <i>Acta Arithmetica</i>. Instytut Matematyczny, 2023. <a href="https://doi.org/10.4064/aa210830-24-11">https://doi.org/10.4064/aa210830-24-11</a>.
  ieee: Y. Diao, “Density of the union of positive diagonal binary quadratic forms,”
    <i>Acta Arithmetica</i>, vol. 207. Instytut Matematyczny, pp. 1–17, 2023.
  ista: Diao Y. 2023. Density of the union of positive diagonal binary quadratic forms.
    Acta Arithmetica. 207, 1–17.
  mla: Diao, Yijie. “Density of the Union of Positive Diagonal Binary Quadratic Forms.”
    <i>Acta Arithmetica</i>, vol. 207, Instytut Matematyczny, 2023, pp. 1–17, doi:<a
    href="https://doi.org/10.4064/aa210830-24-11">10.4064/aa210830-24-11</a>.
  short: Y. Diao, Acta Arithmetica 207 (2023) 1–17.
corr_author: '1'
date_created: 2023-01-26T21:17:04Z
date_published: 2023-01-09T00:00:00Z
date_updated: 2024-10-21T06:01:30Z
day: '09'
department:
- _id: GradSch
doi: 10.4064/aa210830-24-11
external_id:
  arxiv:
  - '2103.08268'
  isi:
  - '000912903000001'
fulldoi: https://doi.org/10.4064/aa210830-24-11
intvolume: '       207'
isi: 1
keyword:
- Algebra
- Number Theory
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2103.08268
month: '01'
oa: 1
oa_version: Preprint
page: 1-17
publication: Acta Arithmetica
publication_identifier:
  eissn:
  - 1730-6264
  issn:
  - 0065-1036
publication_status: published
publisher: Instytut Matematyczny
quality_controlled: '1'
scopus_import: '1'
status: public
title: Density of the union of positive diagonal binary quadratic forms
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 207
year: '2023'
...
---
_id: '17058'
abstract:
- lang: eng
  text: We compare the Manin-type conjecture for Campana points recently formulated
    by Pieropan, Smeets, Tanimoto and Várilly-Alvarado with an alternative prediction
    of Browning and Van Valckenborgh in the special case of the orbifold (P1,D), where
    D=1/2[0]+1/2[1]+1/2[∞]. We find that the two predicted leading constants do not
    agree, and we discuss whether thin sets could explain this discrepancy. Motivated
    by this, we provide a counterexample to the Manin-type conjecture for Campana
    points, by considering orbifolds corresponding to squareful values of binary quadratic
    forms.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Alec L
  full_name: Shute, Alec L
  id: 440EB050-F248-11E8-B48F-1D18A9856A87
  last_name: Shute
  orcid: 0000-0002-1812-2810
citation:
  ama: Shute AL. On the leading constant in the Manin-type conjecture for Campana
    points. <i>Acta Arithmetica</i>. 2022;204(4):317-346. doi:<a href="https://doi.org/10.4064/aa210430-1-7">10.4064/aa210430-1-7</a>
  apa: Shute, A. L. (2022). On the leading constant in the Manin-type conjecture for
    Campana points. <i>Acta Arithmetica</i>. Institute of Mathematics. <a href="https://doi.org/10.4064/aa210430-1-7">https://doi.org/10.4064/aa210430-1-7</a>
  chicago: Shute, Alec L. “On the Leading Constant in the Manin-Type Conjecture for
    Campana Points.” <i>Acta Arithmetica</i>. Institute of Mathematics, 2022. <a href="https://doi.org/10.4064/aa210430-1-7">https://doi.org/10.4064/aa210430-1-7</a>.
  ieee: A. L. Shute, “On the leading constant in the Manin-type conjecture for Campana
    points,” <i>Acta Arithmetica</i>, vol. 204, no. 4. Institute of Mathematics, pp.
    317–346, 2022.
  ista: Shute AL. 2022. On the leading constant in the Manin-type conjecture for Campana
    points. Acta Arithmetica. 204(4), 317–346.
  mla: Shute, Alec L. “On the Leading Constant in the Manin-Type Conjecture for Campana
    Points.” <i>Acta Arithmetica</i>, vol. 204, no. 4, Institute of Mathematics, 2022,
    pp. 317–46, doi:<a href="https://doi.org/10.4064/aa210430-1-7">10.4064/aa210430-1-7</a>.
  short: A.L. Shute, Acta Arithmetica 204 (2022) 317–346.
corr_author: '1'
das_tickbox: '0'
date_created: 2024-05-28T13:39:26Z
date_published: 2022-08-22T00:00:00Z
date_updated: 2026-08-19T12:54:59Z
day: '22'
department:
- _id: TiBr
doi: 10.4064/aa210430-1-7
external_id:
  arxiv:
  - '2104.14946'
  isi:
  - '000844789100001'
fulldoi: https://doi.org/10.4064/aa210430-1-7
intvolume: '       204'
isi: 1
issue: '4'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2104.14946
month: '08'
oa: 1
oa_version: Preprint
page: 317-346
publication: Acta Arithmetica
publication_identifier:
  eissn:
  - 1730-6264
  issn:
  - 0065-1036
publication_status: published
publisher: Institute of Mathematics
quality_controlled: '1'
related_material:
  record:
  - id: '12077'
    relation: earlier_version
    status: public
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: no
title: On the leading constant in the Manin-type conjecture for Campana points
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 204
year: '2022'
...
---
OA_type: closed access
_id: '205'
article_processing_charge: No
article_type: original
author:
- first_name: Timothy D
  full_name: Browning, Timothy D
  id: 35827D50-F248-11E8-B48F-1D18A9856A87
  last_name: Browning
  orcid: 0000-0002-8314-0177
citation:
  ama: Browning TD. Counting rational points on cubic and quartic surfaces. <i>Acta
    Arithmetica</i>. 2003;108:275-295. doi:<a href="https://doi.org/10.4064/aa108-3-7">10.4064/aa108-3-7</a>
  apa: Browning, T. D. (2003). Counting rational points on cubic and quartic surfaces.
    <i>Acta Arithmetica</i>. Instytut Matematyczny. <a href="https://doi.org/10.4064/aa108-3-7">https://doi.org/10.4064/aa108-3-7</a>
  chicago: Browning, Timothy D. “Counting Rational Points on Cubic and Quartic Surfaces.”
    <i>Acta Arithmetica</i>. Instytut Matematyczny, 2003. <a href="https://doi.org/10.4064/aa108-3-7">https://doi.org/10.4064/aa108-3-7</a>.
  ieee: T. D. Browning, “Counting rational points on cubic and quartic surfaces,”
    <i>Acta Arithmetica</i>, vol. 108. Instytut Matematyczny, pp. 275–295, 2003.
  ista: Browning TD. 2003. Counting rational points on cubic and quartic surfaces.
    Acta Arithmetica. 108, 275–295.
  mla: Browning, Timothy D. “Counting Rational Points on Cubic and Quartic Surfaces.”
    <i>Acta Arithmetica</i>, vol. 108, Instytut Matematyczny, 2003, pp. 275–95, doi:<a
    href="https://doi.org/10.4064/aa108-3-7">10.4064/aa108-3-7</a>.
  short: T.D. Browning, Acta Arithmetica 108 (2003) 275–295.
date_created: 2018-12-11T11:45:12Z
date_published: 2003-01-01T00:00:00Z
date_updated: 2026-05-29T07:22:53Z
day: '01'
doi: 10.4064/aa108-3-7
extern: '1'
fulldoi: https://doi.org/10.4064/aa108-3-7
intvolume: '       108'
language:
- iso: eng
month: '01'
oa_version: None
page: 275 - 295
publication: Acta Arithmetica
publication_identifier:
  eissn:
  - 1730-6264
  issn:
  - 0065-1036
publication_status: published
publisher: Instytut Matematyczny
publist_id: '7707'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Counting rational points on cubic and quartic surfaces
type: journal_article
user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd
volume: 108
year: '2003'
...