@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{20423, abstract = {For any d 2, we prove that there exists an integer n0(d) such that there exists an n × n magic square of dth powers for all n n0(d). In particular, we establish the existence of an n × n magic square of squares for all n 4, which settles a conjecture of Várilly-Alvarado. All previous approaches had been based on constructive methods and the existence of n × n magic squares of dth powers had only been known for sparse values of n. We prove our result by the Hardy-Littlewood circle method, which in this setting essentially reduces the problem to finding a sufficient number of disjoint linearly independent subsets of the columns of the coefficient matrix of the equations defining magic squares. We prove an optimal (up to a constant) lower bound for this quantity.}, author = {Rome, Nick and Yamagishi, Shuntaro}, issn = {2363-9555}, journal = {Research in Number Theory}, number = {4}, publisher = {Springer Nature}, title = {{On the existence of magic squares of powers}}, doi = {10.1007/s40993-025-00671-5}, volume = {11}, year = {2025}, } @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{21260, abstract = {We prove that there does not exist F∈Q[x,y] of degree 4 such that F(Z^2 )=Z ≥0. In particular, this answers a question by John S. Lew and Bjorn Poonen for quartic polynomials.}, author = {Yao Xiao, Stanley and Yamagishi, Shuntaro}, issn = {1435-9863}, journal = {Journal of the European Mathematical Society}, publisher = {EMS Press}, title = {{Quartic polynomials in two variables do not represent all non-negative integers}}, doi = {10.4171/jems/1697}, year = {2025}, } @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{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{10018, abstract = {In order to study integral points of bounded log-anticanonical height on weak del Pezzo surfaces, we classify weak del Pezzo pairs. As a representative example, we consider a quartic del Pezzo surface of singularity type A1 + A3 and prove an analogue of Manin's conjecture for integral points with respect to its singularities and its lines.}, author = {Derenthal, Ulrich and Wilsch, Florian Alexander}, issn = {1475-3030 }, journal = {Journal of the Institute of Mathematics of Jussieu}, keywords = {Integral points, del Pezzo surface, universal torsor, Manin’s conjecture}, number = {3}, pages = {1259--1294}, publisher = {Cambridge University Press}, title = {{Integral points on singular del Pezzo surfaces}}, doi = {10.1017/S1474748022000482}, volume = {23}, year = {2024}, } @article{12776, abstract = {An improved asymptotic formula is established for the number of rational points of bounded height on the split smooth del Pezzo surface of degree 5. The proof uses the five conic bundle structures on the surface.}, author = {Browning, Timothy D}, issn = {1076-9803}, journal = {New York Journal of Mathematics}, pages = {1193 -- 1229}, publisher = {State University of New York}, title = {{Revisiting the Manin–Peyre conjecture for the split del Pezzo surface of degree 5}}, volume = {28}, year = {2022}, } @unpublished{10788, abstract = {We determine an asymptotic formula for the number of integral points of bounded height on a certain toric variety, which is incompatible with part of a preprint by Chambert-Loir and Tschinkel. We provide an alternative interpretation of the asymptotic formula we get. To do so, we construct an analogue of Peyre's constant $\alpha$ and describe its relation to a new obstruction to the Zariski density of integral points in certain regions of varieties.}, author = {Wilsch, Florian Alexander}, booktitle = {arXiv}, keywords = {Integral point, toric variety, Manin's conjecture}, title = {{Integral points of bounded height on a certain toric variety}}, doi = {10.48550/arXiv.2202.10909}, year = {2022}, } @article{9199, abstract = {We associate a certain tensor product lattice to any primitive integer lattice and ask about its typical shape. These lattices are related to the tangent bundle of Grassmannians and their study is motivated by Peyre's programme on "freeness" for rational points of bounded height on Fano varieties.}, author = {Browning, Timothy D and Horesh, Tal and Wilsch, Florian Alexander}, issn = {1944-7833}, journal = {Algebra & Number Theory}, number = {10}, pages = {2385--2407}, publisher = {Mathematical Sciences Publishers}, title = {{Equidistribution and freeness on Grassmannians}}, doi = {10.2140/ant.2022.16.2385}, volume = {16}, year = {2022}, } @article{8742, abstract = {We develop a version of Ekedahl’s geometric sieve for integral quadratic forms of rank at least five. As one ranges over the zeros of such quadratic forms, we use the sieve to compute the density of coprime values of polynomials, and furthermore, to address a question about local solubility in families of varieties parameterised by the zeros.}, author = {Browning, Timothy D and Heath-Brown, Roger}, issn = {1435-5337}, journal = {Forum Mathematicum}, number = {1}, pages = {147--165}, publisher = {De Gruyter}, title = {{The geometric sieve for quadrics}}, doi = {10.1515/forum-2020-0074}, volume = {33}, year = {2021}, }