arXiv subjects

Nikolai Bliznyakov

Publications and source records attributed to Nikolai Bliznyakov.

2 recordsLinked to original sources

arXiv2016

Existence of sublattice points in lattice polygons

We state the formula for the critical number of vertices of a convex lattice polygon that guarantees that the polygon contains at least one point of a given sublattice and give a partial proof of the formula. We show that the proof can be reduced to finding upper bounds on the number of vertices in certain classes of polygons. To obtain these bounds, we establish inequalities relating the number of edges of a broken line and the coordinates of its endpoints within a suitable class of broken lines.

math.NT↗

arXiv2016

Bounds on the number of vertices of sublattice-free lattice polygons

In this paper we establish bounds on the number of vertices for a few classes of convex sublattice-free lattice polygons. The bounds are essential for proving the formula for the critical number of vertices of a lattice polygon that ensures the existence of a sublattice point in the polygon. To obtain the bounds, we use relations between the number of edges of lattice broken lines and the coordinates of their endpoints.

math.NT↗