TY - JOUR AB - 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. AU - Browning, Timothy D AU - Hu, L.Q. ID - 6310 JF - Advances in Mathematics SN - 00018708 TI - Counting rational points on biquadratic hypersurfaces VL - 349 ER - TY - JOUR AB - 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. AU - Akopyan, Arseniy AU - Bárány, Imre AU - Robins, Sinai ID - 1180 JF - Advances in Mathematics SN - 00018708 TI - Algebraic vertices of non-convex polyhedra VL - 308 ER -