@article{22929,
  abstract     = {The circle method has been successfully used over the last century to study rational points on hypersurfaces. More recently, a version of the method over function fields, combined with spreading out techniques, has led to a range of results about moduli spaces of rational curves on hypersurfaces. In this paper a version of the circle method is implemented in the setting of the Grothendieck ring of varieties. This allows us to approximate the classes of these moduli spaces directly, without relying on point counting, and leads to a deeper understanding of their geometry.},
  author       = {Bilu, Margaret and Browning, Timothy D},
  issn         = {1873-2151},
  journal      = {Annales Scientifiques de l’École Normale Supérieure},
  keywords     = {Circle method, moduli spaces of curves, hypersurfaces, Grothendieck ring of varieties, motivic integration},
  number       = {5},
  pages        = {1179--1242},
  publisher    = {Société Mathématique de France},
  title        = {{A motivic circle method}},
  doi          = {10.24033/asens.2628},
  volume       = {58},
  year         = {2025},
}

@article{15279,
  abstract     = {We formulate and prove an analog of Poonen’s finite-field Bertini theorem with Taylor conditions that holds in the Grothendieck ring of varieties. This gives a broad generalization of the work of Vakil and Wood, who treated the case of smooth hypersurface sections, and is made possible by the use of motivic Euler products to write down candidate motivic probabilities. As applications, we give motivic analogs of many results in arithmetic statistics that have been proven using Poonen’s sieve, including work of Bucur and Kedlaya on complete intersections and Erman and Wood on semiample Bertini theorems.},
  author       = {Bilu, Margaret and Howe, Sean},
  issn         = {1944-7833},
  journal      = {Algebra & Number Theory},
  keywords     = {Algebra and Number Theory},
  number       = {9},
  pages        = {2195--2259},
  publisher    = {Mathematical Sciences Publishers},
  title        = {{Motivic Euler products in motivic statistics}},
  doi          = {10.2140/ant.2021.15.2195},
  volume       = {15},
  year         = {2021},
}