@article{21002, abstract = {The Davenport–Heilbronn method is a version of the circle method that was developed for studying Diophantine inequalities in the paper (Davenport and Heilbronn, J. Lond. Math. Soc. (1) 21 (1946), 185–193). We discuss the main ideas in the paper, together with an account of the development of the subject in the intervening 80 years.}, author = {Browning, Timothy D}, issn = {1469-7750}, journal = {Journal of the London Mathematical Society}, number = {1}, publisher = {Wiley}, title = {{The Davenport–Heilbronn method: 80 years on}}, doi = {10.1112/jlms.70371}, volume = {113}, year = {2026}, } @article{22618, abstract = {We use a function field version of the circle method to prove that a positive proportion of elements in 𝔽𝑞⁡[𝑡] are representable as a sum of three cubes of minimal degree from 𝔽𝑞⁡[𝑡], assuming a suitable form of the Ratios Conjecture and that char⁡(𝔽𝑞) >3. The analogue of this conjecture for quadratic Dirichlet L-functions is known for large fixed q, via recent developments in homological stability.}, author = {Browning, Timothy D and Glas, Jakob and Wang, Victor}, issn = {2050-5094}, journal = {Forum of Mathematics Sigma}, publisher = {Cambridge University Press}, title = {{Sums of three cubes over a function field}}, doi = {10.1017/fms.2026.10259}, volume = {14}, year = {2026}, } @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{23124, abstract = {Among the set of hypersurfaces of degree d and dimension defined by the vanishing of a homogeneous polynomial with coefficients in the set {−1, +1}, we investigate the probability that a hypersurface contains a rational point as d and tend to ∞.}, author = {Browning, Timothy D and Sawin, Will}, issn = {1464-3847}, journal = {The Quarterly Journal of Mathematics}, publisher = {Oxford University Press}, title = {{Random diophantine equations of large degree}}, doi = {10.1093/qmath/haag037}, year = {2026}, } @article{21244, abstract = {Given a family of varieties over the projective line, we study the density of fibres that are everywhere locally soluble in the case that components of higher multiplicity are allowed. We use log geometry to formulate a new sparsity criterion for the existence of everywhere locally soluble fibres and formulate new conjectures that generalise previous work of Loughran and Smeets. These conjectures involve geometric invariants of the associated multiplicity orbifolds on the base of the fibration in the spirit of Campana. We give evidence for the conjectures by providing an assortment of bounds using Chebotarev’s theorem and sieve methods, with most of the evidence involving upper bounds. }, author = {Browning, Timothy D and Lyczak, Julian and Smeets, Arne}, issn = {1944-7833}, journal = {Algebra & Number Theory}, number = {10}, pages = {2049--2090}, publisher = {Mathematical Sciences Publishers}, title = {{Paucity of rational points on fibrations with multiple fibres}}, doi = {10.2140/ant.2025.19.2049}, volume = {19}, year = {2025}, } @article{21003, abstract = {We extend work of Heath-Brown and Salberger, based on the determinant method, to provide a uniform upper bound for the number of integral points of bounded height on an affine surface, which are subject to a polynomial congruence condition. This is applied to get a new uniform bound for points on diagonal quadric surfaces, and to a problem about the representation of integers as a sum of four unlike powers.}, author = {Browning, Timothy D and Verzobio, Matteo}, issn = {2397-3129}, journal = {Discrete Analysis}, publisher = {Cambridge: Alliance of Diamond Open Access Journals}, title = {{Counting integer points on affine surfaces with a side condition}}, doi = {10.19086/da.143787}, volume = {2025}, year = {2025}, } @article{21266, abstract = {For a given elliptic curve E in short Weierstrass form, we show that almost all quadratic twists E D have no integral points, as D ranges over square-free integers ordered by size. Our result is conditional on a weak form of the Hall–Lang conjecture in the case that E has partial 2-torsion. The proof uses a correspondence of Mordell and the reduction theory of binary quartic forms in order to transfer the problem to counting rational points of bounded height on a certain singular cubic surface, together with extensive use of cancellation in character sum estimates, drawn from Heath-Brown’s analysis of Selmer group statistics for the congruent number curve.}, author = {Browning, Timothy D and Chan, Yik Tung}, issn = {1435-9863}, journal = {Journal of the European Mathematical Society}, publisher = {EMS Press}, title = {{Almost all quadratic twists of an elliptic curve have no integral points}}, doi = {10.4171/jems/1704}, year = {2025}, } @article{19776, abstract = {We use the circle method to prove that a density 1 of elements in Fq[t] are representable as a sum of three cubes of essentially minimal degree from Fq[t], assuming the Ratios Conjecture and that char(Fq)>3. Roughly speaking, to do so, we upgrade an order of magnitude result to a full asymptotic formula that was conjectured by Hooley in the number field setting.}, author = {Browning, Timothy D and Glas, Jakob and Wang, Victor}, issn = {1432-1823}, journal = {Mathematische Zeitschrift}, number = {4}, publisher = {Springer Nature}, title = {{Optimal sums of three cubes in Fq[t]}}, doi = {10.1007/s00209-025-03765-z}, volume = {310}, year = {2025}, } @article{20367, abstract = {We prove upper and lower bounds on the number of pairs of commuting n x n matrices with integer entries in [-T, T], as T -> . Our work uses Fourier analysis and leads to an analysis of exponential sums involving matrices over finite fields. These are bounded by combining a stratification result of Fouvry and Katz with a new result about the flatness of the commutator Lie bracket.}, author = {Browning, Timothy D and Sawin, Will and Wang, Victor}, issn = {1432-1807}, journal = {Mathematische Annalen}, pages = {1863–1880}, publisher = {Springer Nature}, title = {{Pairs of commuting integer matrices}}, doi = {10.1007/s00208-025-03285-5}, volume = {393}, year = {2025}, } @article{21343, abstract = {The large sieve is used to estimate the density of quadratic polynomials Q ∈ Z[x], such that there exists an odd degree polynomial defined over Z which has resultant ±1 with Q. Given a monic polynomial R ∈ Z[x] of odd degree, this is used to show that for almost all quadratic polynomials Q ∈ Z[x], there exists a prime p such that Q and R share a common root in Fp. Using recent work of Landesman, an application to the average size of the odd part of the class group of quadratic number fields is also given}, author = {Browning, Timothy D and Chan, Yik Tung}, issn = {2270-518X}, journal = {Journal de l'Ecole Polytechnique - Mathematiques}, pages = {1677--1691}, publisher = {Ecole Polytechnique}, title = {{Solubility of a resultant equation and applications}}, doi = {10.5802/jep.320}, volume = {12}, year = {2025}, } @article{20249, abstract = {We develop a heuristic for the density of integer points on affine cubic surfaces. Our heuristic applies to smooth surfaces defined by cubic polynomials that are log K3, but it can also be adjusted to handle singular cubic surfaces. We compare our heuristic to Heath-Brown’s prediction for sums of three cubes, as well as to asymptotic formulae in the literature around Zagier’s work on the Markoff cubic surface, and work of Baragar and Umeda on further surfaces of Markoff-type. We also test our heuristic against numerical data for several families of cubic surfaces.}, author = {Browning, Timothy D and Wilsch, Florian Alexander}, issn = {1420-9020}, journal = {Selecta Mathematica New Series}, number = {4}, publisher = {Springer Nature}, title = {{Integral points on cubic surfaces: heuristics and numerics}}, doi = {10.1007/s00029-025-01074-1}, volume = {31}, year = {2025}, } @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{19051, abstract = {This paper corrects an error in an earlier work of the author.}, author = {Browning, Timothy D}, issn = {1687-0247}, journal = {International Mathematics Research Notices}, number = {13}, pages = {10165--10168}, publisher = {Oxford University Press}, title = {{The polynomial sieve and equal sums of like polynomials}}, doi = {10.1093/imrn/rnae066}, volume = {2024}, year = {2024}, } @misc{22234, abstract = {Lists of nontrivial integral points on some (families of) cubic surfaces. (2024-07-18)}, author = {Browning, Timothy D and Sutherland, Andrew}, publisher = {Göttingen Research Online Data}, title = {{Data and code for: Integral points on cubic surfaces: heuristics and numerics}}, doi = {10.25625/4FLFH8}, year = {2024}, } @article{17323, abstract = {We investigate strong divisibility sequences and produce lower and upper bounds for the density of integers in the sequence that only have (somewhat) large prime factors. We focus on the special cases of Fibonacci numbers and elliptic divisibility sequences, discussing the limitations of our methods. At the end of the paper, there is an appendix by Sandro Bettin on divisor closed sets that we use to study the density of prime terms that appear in strong divisibility sequences.}, author = {Browning, Timothy D and Verzobio, Matteo}, issn = {2041-7942}, journal = {Mathematika}, number = {4}, publisher = {London Mathematical Society}, title = {{Strong divisibility sequences and sieve methods}}, doi = {10.1112/mtk.12269}, volume = {70}, year = {2024}, } @article{15338, abstract = {We introduce a new class of generalised quadratic forms over totally real number fields, which is rich enough to capture the arithmetic of arbitrary systems of quadrics over the rational numbers. We explore this connection through a version of the Hardy–Littlewood circle method over number fields.}, author = {Browning, Timothy D and Pierce, Lillian B. and Schindler, Damaris}, issn = {1475-3030}, journal = {Journal of the Institute of Mathematics of Jussieu}, number = {6}, pages = {2859--2912}, publisher = {Cambridge University Press}, title = {{Generalised quadratic forms over totally real number fields}}, doi = {10.1017/S1474748024000161}, volume = {23}, year = {2024}, } @article{15337, abstract = {We prove the Manin–Peyre conjecture for the number of rational points of bounded height outside of a thin subset on a family of Fano threefolds of bidegree (1, 2).}, author = {Bonolis, Dante and Browning, Timothy D and Huang, Zhizhong}, issn = {1432-1807}, journal = {Mathematische Annalen}, pages = {4123--4207}, publisher = {Springer Nature}, title = {{Density of rational points on some quadric bundle threefolds}}, doi = {10.1007/s00208-024-02854-4}, volume = {390}, year = {2024}, } @article{15312, abstract = {The question of whether or not a given integral polynomial takes infinitely many square-free values has only been addressed unconditionally for polynomials of degree at most 3. We address this question, on average, for polynomials of arbitrary degree.}, author = {Browning, Timothy D and Shparlinski, Igor E.}, issn = {0022-314X}, journal = {Journal of Number Theory}, pages = {220--240}, publisher = {Elsevier}, title = {{Square-free values of random polynomials}}, doi = {10.1016/j.jnt.2024.02.013}, volume = {261}, year = {2024}, } @article{13091, abstract = {We use a function field version of the Hardy–Littlewood circle method to study the locus of free rational curves on an arbitrary smooth projective hypersurface of sufficiently low degree. On the one hand this allows us to bound the dimension of the singular locus of the moduli space of rational curves on such hypersurfaces and, on the other hand, it sheds light on Peyre’s reformulation of the Batyrev–Manin conjecture in terms of slopes with respect to the tangent bundle.}, author = {Browning, Timothy D and Sawin, Will}, issn = {1944-7833}, journal = {Algebra & Number Theory}, number = {3}, pages = {719--748}, publisher = {Mathematical Sciences Publishers}, title = {{Free rational curves on low degree hypersurfaces and the circle method}}, doi = {10.2140/ant.2023.17.719}, volume = {17}, year = {2023}, } @article{12916, abstract = {We apply a variant of the square-sieve to produce an upper bound for the number of rational points of bounded height on a family of surfaces that admit a fibration over P1 whose general fibre is a hyperelliptic curve. The implied constant does not depend on the coefficients of the polynomial defining the surface. }, author = {Bonolis, Dante and Browning, Timothy D}, issn = {2036-2145}, journal = {Annali della Scuola Normale Superiore di Pisa, Classe di Scienze}, number = {1}, pages = {173--204}, publisher = {Scuola Normale Superiore - Edizioni della Normale}, title = {{Uniform bounds for rational points on hyperelliptic fibrations}}, doi = {10.2422/2036-2145.202010_018}, volume = {24}, year = {2023}, }