Korkin-Zolotarev bases and successive minima of a lattice and its reciprocal lattice

Let _i(L), _i(L^*) denote the successive minima of a latticeL and its reciprocal latticeL ^*, and let [b₁,..., b _n ] be a basis ofL that is reduced in the sense of Korkin and Zolotarev. We prove that and, where and _j denotes Hermite's constant. As a consequence the inequalities are obtained forn7. Given a basisB of a latticeL in ^m of rankn andx ^m , we define polynomial time computable quantities(B) and(x,B) that are lower bounds for ₁(L) and(x,L), where(x,L) is the Euclidean distance fromx to the closest vector inL. If in additionB is reciprocal to a Korkin-Zolotarev basis ofL ^*, then ₁(L) _n ^* (B) and.The research of the second author was supported by NSF contract DMS 87-06176. The research of the third author was performed at the University of California, Berkeley, with support from NSF grant 21823, and at AT&T Bell Laboratories.     

On integer points in polyhedra: A lower bound

Given a polyhedronP we writeP _I for the convex hull of the integral points inP. It is known thatP _I can have at most135-2 vertices ifP is a rational polyhedron with size . Here we give an example showing thatP _I can have as many as ( ^n–1) vertices. The construction uses the Dirichlet unit theorem.The results of the paper were obtained while this author was visiting the Cowles Foundation at Yale University     

On Vectors whose Span Contains a Given Linear Subspace

Estimates are given for the product of the lengths of integer vectors spanning a given linear subspace.The first author was supported by FWF Austrian Science Fund, project M672.     

On the proof of humbert’s volume formula

We close a gap in Humbert’s classical calculation of the volume of the quotient of three-dimensional hyperbolic space by SL₂ over the ring of integers of an imaginary quadratic number field.     

Counting points of fixed degree and bounded height on linear varieties

We count points of fixed degree and bounded height on a linear projective variety defined over a number field k. If the dimension of the variety is large enough compared to the degree we derive asymptotic estimates as the height tends to infinity. This generalizes results of Thunder, Christensen and Gubler and special cases of results of Schmidt and Gao.     

On subgroups of minimal topological groups

A topological group is minimal if it does not admit a strictly coarser Hausdorff group topology. The Roelcke uniformity (or lower uniformity) on a topological group is the greatest lower bound of the left and right uniformities. A group is Roelcke-precompact if it is precompact with respect to the Roelcke uniformity. Many naturally arising non-Abelian topological groups are Roelcke-precompact and hence have a natural compactification. We use such compactifications to prove that some groups of isometries are minimal. In particular, if U₁ is the Urysohn universal metric space of diameter 1, the group Iso(U₁) of all self-isometries of U₁ is Roelcke-precompact, topologically simple and minimal. We also show that every topological group is a subgroup of a minimal topologically simple Roelcke-precompact group of the form Iso(M), where M is an appropriate non-separable version of the Urysohn space.     

Canonical height functions for affine plane automorphisms

Let be a polynomial automorphism of dynamical degree δ≥2 over a number field K. We construct height functions defined on that transform well relative to f, which we call canonical height functions for f. These functions satisfy the Northcott finiteness property, and a -valued point on is f-periodic if and only if its height is zero. As an application, we give an estimate on the number of points with bounded height in an infinite f-orbit.     

On the number of nowhere zero points in linear mappings

LetA be a nonsingularn byn matrix over the finite fieldGF _q ,k=n/2,q=p ^a ,a1, wherep is prime. LetP(A,q) denote the number of vectorsx in (GF _q ) ⁿ such that bothx andAx have no zero component. We prove that forn2, and ,P(A,q)[(q–1)(q–3)] ^k (q–2) ^n–2k and describe all matricesA for which the equality holds. We also prove that the result conjectured in [1], namely thatP(A,q)1, is true for allqn+23 orqn+14.     

On the proof of Humbert's volume formula

We close a gap in Humbert's classical calculation of the volume of the quotient of three-dimensional hyperbolic space by SL² over the ring of integers of an imaginary quadratic number field. Received: 19 December 1996 / Revised version: 4 September 1997     

On odd points and the volume of lattice polyhedra

Denote by _n ³ ,n 2, the lattice consisting of all pointsx in ³ such thatnx belongs to the fundamental lattice ³ of points with integer coordinates. Letl _n be the subset of _n ³ consisting of all points whose coordinates are odd multiples of 1/n. The purpose of this paper is to give several new Pick-type formulae for the volume of three-dimensional lattice polyhedra, that is, polyhedra with vertices in ³. Our formulae are in terms of numbers of only thel _n-points belonging to a lattice polyhedronP in contrast to already known formulae which employ numbers of all the _n ³ -points inP. On our way to establishing the formulae we show that the number of points froml _n belonging to a three-dimensional lattice polyhedronP has some polynomiality properties similar to those of the well-known Ehrhart polynomial expressing the number of points of _n ³ inP. The paper contains also some comments on a problem of finding a volume formula which would employ only the setsl _n and which would be applicable to lattice polyhedra in arbitrary dimensions.Research partially supported by KBN Grant 2 P03A 008 10.     

On the Dirichlet—Voronoi cell of unimodular lattices

This paper consists of two results concerning the Dirichlet-Voronoi cell of a lattice. The first one is a geometric property of the cell of an integral unimodular lattice while the second one gives a characterization of all those lattice vectors of an arbitrary lattice whose multiples by 1/2 are on the boundary of the cell containing the origin. This result is a generalization of a well-known theorem of Voronoi characterizing the so-called relevants of the cell.Supported by Hung. Nat. Found. for Sci. Research (OTKA) grant No. T.7351 (1994).     

On the density of the minimal subspaces generated by discrete linear Hamiltonian systems

This paper focuses on the density of the minimal subspaces generated by a class of discrete linear Hamiltonian systems. It is shown that the minimal subspace is densely defined if and only if the maximal subspace is an operator; that is, it is single valued. In addition, it is shown that, if the interval on which the systems are defined is bounded from below or above, then the minimal subspace is non-densely defined in any non-trivial case.     

On a minimal factorization conjecture

Let be a proper holomorphic map from a connected complex surface S onto the open unit disk D⊂C, with 0∈D as its unique singular value, and having fiber genus g>0. Assume that in case g?2, admits a deformation whose singular fibers are all of simple Lefschetz type. It has been conjectured that the factorization of the monodromy f∈Mg around ?⁻¹(0) in terms of right-handed Dehn twists induced by the monodromy of has the least number of factors among all possible factorizations of f as a product of right-handed Dehn twists in the mapping class group (see [M. Ishizaka, One parameter families of Riemann surfaces and presentations of elements of mapping class group by Dehn twists, J. Math. Soc. Japan 58 (2) (2006) 585-594]). In this article, the validity of this conjecture is established for g=1.     

Computability of finite-dimensional linear subspaces and best approximation

We discuss computability properties of the set P_G(x) of elements of best approximation of some point x∈X by elements of G⊆X in computable Banach spaces X. It turns out that for a general closed set G, given by its distance function, we can only obtain negative information about P_G(x) as a closed set. In the case that G is finite-dimensional, one can compute negative information on P_G(x) as a compact set. This implies that one can compute the point in P_G(x) whenever it is uniquely determined. This is also possible for a wider class of subsets G, given that one imposes additionally convexity properties on the space. If the Banach space X is computably uniformly convex and G is convex, then one can compute the uniquely determined point in P_G(x). We also discuss representations of finite-dimensional subspaces of Banach spaces and we show that a basis representation contains the same information as the representation via distance functions enriched by the dimension. Finally, we study computability properties of the dimension and the codimension map and we show that for finite-dimensional spaces X the dimension is computable, given the distance function of the subspace.     

On the residue classes of integer sequences satisfying a linear three term recurrence formula

In this paper we investigate linear three-term recurrence formulae with sequences of integers (T(n))_n?0 and (U(n))_n?0, which are ultimately periodic modulo m, e.g.

     

On the <Emphasis Type="Italic">DOTU</Emphasis>-Factorization Problem in Dimension Four

 We prove that any basis of a non-degenerate 4-dimensional lattice with sufficiently small (positive) homogeneous minimum can be represented in the form DOTU. This is of interest in connection with Minkowski’s conjecture about the product of inhomogeneous linear forms. Received 23 September 2001 RID="a" ID="a" Dedicated to Prof. Edmund Hlawka on the occasion of his 85th birthday