@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},
}