@article{22821,
  abstract     = {With probability 1, we assess the average behaviour of various arithmetic functions at the values of degree 𝑑 polynomials 𝑓∈ℤ⁢[𝑡] that are ordered by height. This allows us to establish averaged versions of the Bateman–Horn conjecture, the polynomial Chowla conjecture and to address a basic question about the integral Hasse principle for norm form equations. Moreover, we are able to quantify the error term in the asymptotics and the size of the exceptional set of 𝑓, both with arbitrary logarithmic power savings.},
  author       = {Browning, Timothy D and Sofos, Efthymios and Teräväinen, Joni},
  issn         = {1435-5345},
  journal      = {Journal für die reine und angewandte Mathematik},
  publisher    = {De Gruyter},
  title        = {{Bateman-Horn, polynomial Chowla and the Hasse principle with probability}},
  doi          = {10.1515/crelle-2026-0062},
  year         = {2026},
}

@article{256,
  abstract     = {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.},
  author       = {Browning, Timothy D and Prendiville, Sean},
  issn         = {0075-4102},
  journal      = {Journal fur die Reine und Angewandte Mathematik},
  number       = {731},
  pages        = {122},
  publisher    = {Walter de Gruyter},
  title        = {{Improvements in Birch's theorem on forms in many variables}},
  doi          = {10.1515/crelle-2014-0122},
  volume       = {2017},
  year         = {2017},
}

@article{271,
  abstract     = {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.},
  author       = {Browning, Timothy D and Prendiville, Sean},
  issn         = {0075-4102},
  journal      = {Journal fur die Reine und Angewandte Mathematik},
  number       = {731},
  pages        = {203 -- 234},
  publisher    = {Walter de Gruyter},
  title        = {{Improvements in Birch's theorem on forms in many variables}},
  doi          = {10.1515/crelle-2014-0122},
  volume       = {2017},
  year         = {2015},
}

@article{212,
  abstract     = {For any n ≧ 2, let F ∈ ℤ [ x 1, … , xn ] be a form of degree d≧ 2, which produces a geometrically irreducible hypersurface in ℙn–1. This paper is concerned with the number N(F;B) of rational points on F = 0 which have height at most B. For any ε > 0 we establish the estimate N(F; B) = O(B n− 2+ ε ), whenever either n ≦ 5 or the hypersurface is not a union of lines. Here the implied constant depends at most upon d, n and ε.},
  author       = {Browning, Timothy D and Heath Brown, Roger},
  issn         = {0075-4102},
  journal      = {Journal fur die Reine und Angewandte Mathematik},
  number       = {584},
  pages        = {83 -- 115},
  publisher    = {Walter de Gruyter and Co },
  title        = {{Counting rational points on hypersurfaces}},
  doi          = {https://doi.org/10.1515/crll.2005.2005.584.83},
  year         = {2005},
}