Covering Radius of Two-dimensional Lattices

The covering radius problem in any dimension is not known to be solvable in nondeterministic polynomial time, but when in dimension two, we give a deterministic polynomial time algorithm by computing a reduced basis using Gauss\u27 algorithm in this paper

On lattice extensions

A lattice $\Lambda$ is said to be an extension of a sublattice $L$ of smaller rank if $L$ is equal to the intersection of $\Lambda$ with the subspace spanned by $L$. The goal of this paper is to initiate a systematic study of the geometry of lattice extensions. We start by proving the existence of a small-determinant extension of a given lattice, and then look at successive minima and covering radius. To this end, we investigate extensions (within an ambient lattice) preserving the successive minima of the given lattice, as well as extensions preserving the covering radius. We also exhibit some interesting arithmetic properties of deep holes of planar lattices.Comment: 18 pages, 4 figures; to appear in Monatshefte f\"ur Mathemati

Lattice Extensions and Zeros of Multilinear Polynomials

We treat several problems related to the existence of lattice extensions preserving certain geometric properties and small-height zeros of various multilinear polynomials. An extension of a Euclidean lattice $L_1$ is a lattice $L_2$ of higher rank containing $L_1$ so that the intersection of $L_2$ with the subspace spanned by $L_1$ is equal to $L_1$. Our first result provides a counting estimate on the number of ways a primitive collection of vectors in a lattice can be extended to a basis for this lattice. Next, we discuss the existence of lattice extensions with controlled determinant, successive minima and covering radius. In the two-dimensional case, we also present some observations about the deep holes of a lattice as elements of the quotient torus group. Looking for basis extensions additionally connects to a search for small-height zeros of multilinear polynomials, for which we obtain several results over arbitrary number fields. These include bounds for a system of polynomials under appropriate hypotheses, as well as for a single polynomial with some additional avoidance conditions. In addition to several height inequalities that we need for these bounds, we obtain a new absolute version of Siegel\u27s lemma which is proved using only linear algebra tools