Related papers: Density of integral points on algebraic varieties

We estimate the maximal number of integral points which can be on a convex arc in the plane with given length, minimal radius of curvature and initial slope.

We initiate a systematic study of the deep points of a cluster algebra; that is, the points in the associated variety which are not in any cluster torus. We describe the deep points of cluster algebras of type A, rank 2, Markov, and…

We prove some general density statements about the subgroup of invertible points on intermediate jacobians; namely those points in the Abel-Jacobi image of nullhomologous algebraic cycles on projective algebraic manifolds.

Generalized diffusion type equations are considered and point symmetry analysis is applied to them. The equations with extremal order point symmetry algebras are described. Some old geometrical results are rederived in connection with…

It is investigated how graded variants of integral and complete integral closures behave under coarsening functors and under formation of group algebras.

We establish a version of a semistable reduction theorem over a log point with a non-trivial nilpotent structure. In order to do this we extend the classical desingularization theories to non-reduced schemes with generically principal…

We characterise and investigate co-Higgs sheaves and associated algebraic and combinatorial invariants on toric varieties. In particular, we compute explicit examples.

We aim at studying collections of algebraic structures defined over a commutative ring and investigating the complexity of significant constructions carried out on these objects. The assignment of measures of size, via a multiplicity…

We establish sharp lower and upper bounds for the number of integral points near dilations of a space curve with nowhere vanishing torsion.

We consider some variations on the classical method of Runge for effectively determining integral points on certain curves. We first prove a version of Runge's theorem valid for higher-dimensional varieties, generalizing a uniform version…

We study the multi-height distribution of rational points of smooth, projective and split toric varieties over $\mathbf{Q}$ using the lift of the number of points to universal torsors.

These notes provide an overview of various notions of hyperbolicity for varieties of log general type from the viewpoint of both arithmetic and birational geometry. The main results are based on our paper entitled "Hyperbolicity and…

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…

In this paper, we provide evidence to support a positive answer to a question of Hassett and Tschinkel. In particular, if an algebraic variety V has a dense set of rational points, they ask whether or not the set of D-integral points is…

We consider families of smooth projective curves of genus 2 with a single point removed and study their integral points. We show that in many such families there is a dense set of fibres for which the integral points can be effectively…

In this short note, we describe the so-called homogeneous involution on finite-dimensional graded-division algebra over an algebraically closed field. We also compute their graded polynomial identities with involution. As pointed out by L.…

Grassmann and flag varieties lead many lives in pure and applied mathematics. Here we focus on the algebraic complexity of solving various problems in linear algebra and statistics as optimization problems over these varieties. The measure…

We discuss different generalizations of the classical notion of the index of a singular point of a vector field to the case of vector fields or 1-forms on singular varieties, describe relations between them and formulae for their…

Following a purely algebraic procedure, we provide an exhaustive classification of local Weyl-invariant scalar densities in dimension D=8.

For affine algebraic plane curves we reduce a calculation of its invariants to calculation of the intersection of kernels of some derivations.