---
_id: '11636'
abstract:
- lang: eng
  text: In [3], Poonen and Slavov recently developed a novel approach to Bertini irreducibility
    theorems over an arbitrary field, based on random hyperplane slicing. In this
    paper, we extend their work by proving an analogous bound for the dimension of
    the exceptional locus in the setting of linear subspaces of higher codimensions.
article_number: '102085'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Philip
  full_name: Kmentt, Philip
  id: c90670c9-0bf0-11ed-86f5-ed522ece2fac
  last_name: Kmentt
- 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: Kmentt P, Shute AL. The Bertini irreducibility theorem for higher codimensional
    slices. <i>Finite Fields and their Applications</i>. 2022;83(10). doi:<a href="https://doi.org/10.1016/j.ffa.2022.102085">10.1016/j.ffa.2022.102085</a>
  apa: Kmentt, P., &#38; Shute, A. L. (2022). The Bertini irreducibility theorem for
    higher codimensional slices. <i>Finite Fields and Their Applications</i>. Elsevier.
    <a href="https://doi.org/10.1016/j.ffa.2022.102085">https://doi.org/10.1016/j.ffa.2022.102085</a>
  chicago: Kmentt, Philip, and Alec L Shute. “The Bertini Irreducibility Theorem for
    Higher Codimensional Slices.” <i>Finite Fields and Their Applications</i>. Elsevier,
    2022. <a href="https://doi.org/10.1016/j.ffa.2022.102085">https://doi.org/10.1016/j.ffa.2022.102085</a>.
  ieee: P. Kmentt and A. L. Shute, “The Bertini irreducibility theorem for higher
    codimensional slices,” <i>Finite Fields and their Applications</i>, vol. 83, no.
    10. Elsevier, 2022.
  ista: Kmentt P, Shute AL. 2022. The Bertini irreducibility theorem for higher codimensional
    slices. Finite Fields and their Applications. 83(10), 102085.
  mla: Kmentt, Philip, and Alec L. Shute. “The Bertini Irreducibility Theorem for
    Higher Codimensional Slices.” <i>Finite Fields and Their Applications</i>, vol.
    83, no. 10, 102085, Elsevier, 2022, doi:<a href="https://doi.org/10.1016/j.ffa.2022.102085">10.1016/j.ffa.2022.102085</a>.
  short: P. Kmentt, A.L. Shute, Finite Fields and Their Applications 83 (2022).
date_created: 2022-07-24T22:01:41Z
date_published: 2022-10-01T00:00:00Z
date_updated: 2023-08-03T12:12:57Z
day: '01'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.1016/j.ffa.2022.102085
external_id:
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publication: Finite Fields and their Applications
publication_identifier:
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publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: The Bertini irreducibility theorem for higher codimensional slices
tmp:
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  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
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volume: 83
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...
---
_id: '12072'
abstract:
- lang: eng
  text: "In this thesis, we study two of the most important questions in Arithmetic
    geometry: that of the existence and density of solutions to Diophantine equations.
    In order for a Diophantine equation to have any solutions over the rational numbers,
    it must have solutions everywhere locally, i.e., over R and over Qp for every
    prime p. The converse, called the Hasse principle, is known to fail in general.
    However, it is still a central question in Arithmetic geometry to determine for
    which varieties the Hasse principle does hold. In this work, we establish the
    Hasse principle for a wide new family of varieties of the form f(t) = NK/Q(x)
    ̸= 0, where f is a polynomial with integer coefficients and NK/Q denotes the norm\r\nform
    associated to a number field K. Our results cover products of arbitrarily many
    linear, quadratic or cubic factors, and generalise an argument of Irving [69],
    which makes use of the beta sieve of Rosser and Iwaniec. We also demonstrate how
    our main sieve results can be applied to treat new cases of a conjecture of Harpaz
    and Wittenberg on locally split values of polynomials over number fields, and
    discuss consequences for rational points in fibrations.\r\nIn the second question,
    about the density of solutions, one defines a height function and seeks to estimate
    asymptotically the number of points of height bounded by B as B → ∞. Traditionally,
    one either counts rational points, or\r\nintegral points with respect to a suitable
    model. However, in this thesis, we study an emerging area of interest in Arithmetic
    geometry known as Campana points, which in some sense interpolate between rational
    and integral points.\r\nMore precisely, we count the number of nonzero integers
    z1, z2, z3 such that gcd(z1, z2, z3) = 1, and z1, z2, z3, z1 + z2 + z3 are all
    squareful and bounded by B. Using the circle method, we obtain an asymptotic formula
    which agrees in\r\nthe power of B and log B with a bold new generalisation of
    Manin’s conjecture to the setting of Campana points, recently formulated by Pieropan,
    Smeets, Tanimoto and Várilly-Alvarado [96]. However, in this thesis we also provide
    the first known counterexamples to leading constant predicted by their conjecture. "
acknowledgement: I acknowledge the received funding from the European Union’s Horizon
  2020 research and innovation programme under the Marie Sklodowska Curie Grant Agreement
  No. 665385.
alternative_title:
- ISTA Thesis
article_processing_charge: No
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. Existence and density problems in Diophantine geometry: From norm
    forms to Campana points. 2022. doi:<a href="https://doi.org/10.15479/at:ista:12072">10.15479/at:ista:12072</a>'
  apa: 'Shute, A. L. (2022). <i>Existence and density problems in Diophantine geometry:
    From norm forms to Campana points</i>. Institute of Science and Technology Austria.
    <a href="https://doi.org/10.15479/at:ista:12072">https://doi.org/10.15479/at:ista:12072</a>'
  chicago: 'Shute, Alec L. “Existence and Density Problems in Diophantine Geometry:
    From Norm Forms to Campana Points.” Institute of Science and Technology Austria,
    2022. <a href="https://doi.org/10.15479/at:ista:12072">https://doi.org/10.15479/at:ista:12072</a>.'
  ieee: 'A. L. Shute, “Existence and density problems in Diophantine geometry: From
    norm forms to Campana points,” Institute of Science and Technology Austria, 2022.'
  ista: 'Shute AL. 2022. Existence and density problems in Diophantine geometry: From
    norm forms to Campana points. Institute of Science and Technology Austria.'
  mla: 'Shute, Alec L. <i>Existence and Density Problems in Diophantine Geometry:
    From Norm Forms to Campana Points</i>. Institute of Science and Technology Austria,
    2022, doi:<a href="https://doi.org/10.15479/at:ista:12072">10.15479/at:ista:12072</a>.'
  short: 'A.L. Shute, Existence and Density Problems in Diophantine Geometry: From
    Norm Forms to Campana Points, Institute of Science and Technology Austria, 2022.'
date_created: 2022-09-08T21:53:03Z
date_published: 2022-09-08T00:00:00Z
date_updated: 2023-02-21T16:37:35Z
day: '08'
ddc:
- '512'
degree_awarded: PhD
department:
- _id: GradSch
- _id: TiBr
doi: 10.15479/at:ista:12072
ec_funded: 1
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has_accepted_license: '1'
language:
- iso: eng
license: https://creativecommons.org/licenses/by-nc-sa/4.0/
month: '09'
oa: 1
oa_version: Published Version
page: '208'
project:
- _id: 2564DBCA-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '665385'
  name: International IST Doctoral Program
publication_identifier:
  isbn:
  - 978-3-99078-023-7
  issn:
  - 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
related_material:
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    relation: part_of_dissertation
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  - id: '12077'
    relation: part_of_dissertation
    status: public
status: public
supervisor:
- first_name: Timothy D
  full_name: Browning, Timothy D
  id: 35827D50-F248-11E8-B48F-1D18A9856A87
  last_name: Browning
  orcid: 0000-0002-8314-0177
title: 'Existence and density problems in Diophantine geometry: From norm forms to
  Campana points'
tmp:
  image: /images/cc_by_nc_sa.png
  legal_code_url: https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode
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    BY-NC-SA 4.0)
  short: CC BY-NC-SA (4.0)
type: dissertation
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2022'
...
---
_id: '12076'
abstract:
- lang: eng
  text: We find an asymptotic formula for the number of primitive vectors $(z_1,\ldots,z_4)\in
    (\mathbb{Z}_{\neq 0})^4$ such that $z_1,\ldots, z_4$ are all squareful and bounded
    by $B$, and $z_1+\cdots + z_4 = 0$. Our result agrees in the power of $B$ and
    $\log B$ with the Campana-Manin conjecture of Pieropan, Smeets, Tanimoto and V\'{a}rilly-Alvarado.
article_number: '2104.06966'
article_processing_charge: No
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. Sums of four squareful numbers. <i>arXiv</i>. doi:<a href="https://doi.org/10.48550/arXiv.2104.06966">10.48550/arXiv.2104.06966</a>
  apa: Shute, A. L. (n.d.). Sums of four squareful numbers. <i>arXiv</i>. <a href="https://doi.org/10.48550/arXiv.2104.06966">https://doi.org/10.48550/arXiv.2104.06966</a>
  chicago: Shute, Alec L. “Sums of Four Squareful Numbers.” <i>ArXiv</i>, n.d. <a
    href="https://doi.org/10.48550/arXiv.2104.06966">https://doi.org/10.48550/arXiv.2104.06966</a>.
  ieee: A. L. Shute, “Sums of four squareful numbers,” <i>arXiv</i>. .
  ista: Shute AL. Sums of four squareful numbers. arXiv, 2104.06966.
  mla: Shute, Alec L. “Sums of Four Squareful Numbers.” <i>ArXiv</i>, 2104.06966,
    doi:<a href="https://doi.org/10.48550/arXiv.2104.06966">10.48550/arXiv.2104.06966</a>.
  short: A.L. Shute, ArXiv (n.d.).
date_created: 2022-09-09T10:42:51Z
date_published: 2021-04-15T00:00:00Z
date_updated: 2023-02-21T16:37:30Z
day: '15'
department:
- _id: TiBr
doi: 10.48550/arXiv.2104.06966
external_id:
  arxiv:
  - '2104.06966'
language:
- iso: eng
main_file_link:
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  url: https://doi.org/10.48550/arXiv.2104.06966
month: '04'
oa: 1
oa_version: Preprint
publication: arXiv
publication_status: submitted
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status: public
title: Sums of four squareful numbers
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2021'
...
---
_id: '12077'
abstract:
- lang: eng
  text: "We compare the Manin-type conjecture for Campana points recently formulated\r\nby
    Pieropan, Smeets, Tanimoto and V\\'{a}rilly-Alvarado with an alternative\r\nprediction
    of Browning and Van Valckenborgh in the special case of the orbifold\r\n$(\\mathbb{P}^1,D)$,
    where $D =\\frac{1}{2}[0]+\\frac{1}{2}[1]+\\frac{1}{2}[\\infty]$. We find that
    the two predicted leading constants do not agree, and we discuss whether thin
    sets\r\ncould explain this discrepancy. Motivated by this, we provide a counterexample\r\nto
    the Manin-type conjecture for Campana points, by considering orbifolds\r\ncorresponding
    to squareful values of binary quadratic forms."
acknowledgement: The author would like to thank Damaris Schindler and Florian Wilsch
  for their helpful comments on the heights and Tamagawa measures used in Section
  3, together with Marta Pieropan, Sho Tanimoto and Sam Streeter for providing valuable
  feedback on an earlier version of this paper, and Tim Browning for many useful comments
  and discussions during the development of this work. The author is also grateful
  to the anonymous referee for providing many valuable comments and suggestions that
  improved the quality of the paper.
article_number: '2104.14946'
article_processing_charge: No
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>arXiv</i>. doi:<a href="https://doi.org/10.48550/arXiv.2104.14946">10.48550/arXiv.2104.14946</a>
  apa: Shute, A. L. (n.d.). On the leading constant in the Manin-type conjecture for
    Campana points. <i>arXiv</i>. <a href="https://doi.org/10.48550/arXiv.2104.14946">https://doi.org/10.48550/arXiv.2104.14946</a>
  chicago: Shute, Alec L. “On the Leading Constant in the Manin-Type Conjecture for
    Campana Points.” <i>ArXiv</i>, n.d. <a href="https://doi.org/10.48550/arXiv.2104.14946">https://doi.org/10.48550/arXiv.2104.14946</a>.
  ieee: A. L. Shute, “On the leading constant in the Manin-type conjecture for Campana
    points,” <i>arXiv</i>. .
  ista: Shute AL. On the leading constant in the Manin-type conjecture for Campana
    points. arXiv, 2104.14946.
  mla: Shute, Alec L. “On the Leading Constant in the Manin-Type Conjecture for Campana
    Points.” <i>ArXiv</i>, 2104.14946, doi:<a href="https://doi.org/10.48550/arXiv.2104.14946">10.48550/arXiv.2104.14946</a>.
  short: A.L. Shute, ArXiv (n.d.).
date_created: 2022-09-09T10:43:17Z
date_published: 2021-04-30T00:00:00Z
date_updated: 2023-02-21T16:37:30Z
day: '30'
department:
- _id: TiBr
doi: 10.48550/arXiv.2104.14946
external_id:
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  - '2104.14946'
language:
- iso: eng
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month: '04'
oa: 1
oa_version: Preprint
publication: arXiv
publication_status: submitted
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title: On the leading constant in the Manin-type conjecture for Campana points
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2021'
...