@article{10765,
abstract = {We establish the Hardy-Littlewood property (à la Borovoi-Rudnick) for Zariski open subsets in affine quadrics of the form q(x1,...,xn)=m, where q is a non-degenerate integral quadratic form in n>3 variables and m is a non-zero integer. This gives asymptotic formulas for the density of integral points taking coprime polynomial values, which is a quantitative version of the arithmetic purity of strong approximation property off infinity for affine quadrics.},
author = {Cao, Yang and Huang, Zhizhong},
issn = {10902082},
journal = {Advances in Mathematics},
number = {3},
publisher = {Elsevier},
title = {{Arithmetic purity of the Hardy-Littlewood property and geometric sieve for affine quadrics}},
doi = {10.1016/j.aim.2022.108236},
volume = {398},
year = {2022},
}
@article{6310,
abstract = {An asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain Zariskiopen subset of an arbitrary smooth biquadratic hypersurface in sufficiently many variables. The proof uses the Hardy–Littlewood circle method.},
author = {Browning, Timothy D and Hu, L.Q.},
issn = {10902082},
journal = {Advances in Mathematics},
pages = {920--940},
publisher = {Elsevier},
title = {{Counting rational points on biquadratic hypersurfaces}},
doi = {10.1016/j.aim.2019.04.031},
volume = {349},
year = {2019},
}
@article{1180,
abstract = {In this article we define an algebraic vertex of a generalized polyhedron and show that the set of algebraic vertices is the smallest set of points needed to define the polyhedron. We prove that the indicator function of a generalized polytope P is a linear combination of indicator functions of simplices whose vertices are algebraic vertices of P. We also show that the indicator function of any generalized polyhedron is a linear combination, with integer coefficients, of indicator functions of cones with apices at algebraic vertices and line-cones. The concept of an algebraic vertex is closely related to the Fourier–Laplace transform. We show that a point v is an algebraic vertex of a generalized polyhedron P if and only if the tangent cone of P, at v, has non-zero Fourier–Laplace transform.},
author = {Akopyan, Arseniy and Bárány, Imre and Robins, Sinai},
issn = {00018708},
journal = {Advances in Mathematics},
pages = {627 -- 644},
publisher = {Academic Press},
title = {{Algebraic vertices of non-convex polyhedra}},
doi = {10.1016/j.aim.2016.12.026},
volume = {308},
year = {2017},
}