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.     

A Property of Polynomials with an Applicationto Siegel’s Lemma

 It is proved that natural necessary conditions imply the existence of infinitely many integer points at which given multivariate polynomials with integer coefficients take coprime values. As a consequence the best constant in the simplest case of Siegel’s lemma is expressed in terms of critical determinants of suitable star bodies. Received August 10, 2001; in revised form March 13, 2002 RID="a" ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday     

A Lower Bound for the Height of a Rational Function at <Emphasis Type="Italic">S</Emphasis>-unit Points

Let a,b be given, multiplicatively independent positive integers and let >0. In a recent paper jointly with Y. Bugeaud we proved the upper bound exp(n) for g.c.d.(aⁿ–1, bⁿ–1); shortly afterwards we generalized this to the estimate g.c.d.(u–1,v–1)v) for multiplicatively independent S-units u,vZ. In a subsequent analysis of those results it turned out that a perhaps better formulation of them may be obtained in terms of the language of heights of algebraic numbers. In fact, the purposes of the present paper are: to generalize the upper bound for the g.c.d. to pairs of rational functions other than {u–1,v–1} and to extend the results to the realm of algebraic numbers, giving at the same time a new formulation of the bounds in terms of height functions and algebraic subgroups of G_m².     

Linear Spaces on the Intersection of Two Quadratic Hypersurfaces,and Systems of <Emphasis Type="Italic">p</Emphasis>-adic Quadratic Forms

We obtain new results on linear spaces on the intersection of two quadratic forms defined over a non-dyadic p-adic field . One of our main tools is a recent result of Parimala and Suresh on isotropy of quadratic forms over functions fields over . As a corollary we also get new bounds for the number of variables necessary to always find a non-trivial p-adic zero of a system of quadratic forms.     

Approximation d’un nombre réel par des unités algébriques

 In this paper, we prove that for any real number ξ, which is not an algebraic number of degree , there exist infinitely many real algebraic units α of degree n + 1 such that . We also show how the flexibility of H. Davenport and W. M. Schmidt’s method allows to replace, with the same exponent of approximation, units of degree over Z (i.e. elements α with both α and integral over Z) by units of degree over a finite intersection .

(Received 14 March 2000; in revised form 16 November 2000)     

Cycles of Polynomial Mappings in Several Variables over Rings of Integers in Finite Extensions of the Rationals,II

We prove (Theorem 1.1) an asymptotic for maximal cycle lengths of polynomial mappings in several variables over rings of integers in algebraic number fields. In addition (Proposition 3.1), we strengthen the shape of possible lengths of polynomial cycles over discrete valuation domains.     

Forms Represented by Linear Forms

This paper is the third in a series in which the author investigates the question of representation of forms by linear forms. Whereas in the first two treatments the proportion of forms F of degree 3 (resp. degree d) which can be written as a sum of two cubes (resp. d-th powers) of linear forms with algebraic coefficients is determined, the generalization now consists in allowing more general expressions of degree d in two linear forms. The main result is thus to give an asymptotic formula, in terms of their height, for the number or decomposable forms that have a representation

Almost Perfect Powers in Products of Consecutive Integers

The classical equations (1) and (3) have a very extensive literature. The main purpose of recent investigations is to solve these equations with as large bound for the greatest prime factor P(b) of b as possible. General elementary methods have been developed for studying (1) and (3) which, however, cannot be applied if k is small. As a generalization of previous results obtained for small values of k, we completely solve Eq. (1) for k 5, under the assumption that P(b) p_k, the k-th prime (cf. Theorem 1). A similar result is established for Eq. (3) (cf. Theorem 2). In our proofs, several deep results and powerful techniques are combined from modern Diophantine analysis.     

Counting Singular Matrices with Primitive Row Vectors

We solve an asymptotic problem in the geometry of numbers, where we count the number of singular n×n matrices where row vectors are primitive and of length at most T. Without the constraint of primitivity, the problem was solved by Y. Katznelson. We show that as T, the number is asymptotic to for n3. The 3-dimensional case is the most problematic and we need to invoke an equidistribution theorem due to W. M. Schmidt.     

Irreducibility Results for Compositions of Polynomials with Integer Coefficients

We obtain explicit upper bounds for the number of irreducible factors for a class of polynomials of the form f ○ g, where f,g are polynomials with integer coefficients, in terms of the prime factorization of the leading coefficients of f and g, the degrees of f and g, and the size of coefficients of f. In particular, some irreducibility results are given for this class of compositions of polynomials.     

On the Packing Densities and the Covering Densities of the Cartesian Products of Convex Bodies

In this article we study the following problem: Is the covering (packing) density of a Cartesian product of two convex bodies always equal to the product of their corresponding covering (packing) densities? For the covering case we get a negative answer. For the packing case we get a combinatorial version which seems to be important for its own interest.     

Dyadic Diaphony of Digital Nets Over ℤ<Subscript>2</Subscript>

The dyadic diaphony, introduced by Hellekalek and Leeb, is a quantitative measure for the irregularity of distribution of point sets in the unit-cube. In this paper we study the dyadic diaphony of digital nets over ℤ₂. We prove an upper bound for the dyadic diaphony of nets and show that the convergence order is best possible. This follows from a relation between the dyadic diaphony and the discrepancy. In order to investigate the case where the number of points is small compared to the dimension we introduce the limiting dyadic diaphony, which is defined as the limiting case where the dimension tends to infinity. We obtain a tight upper and lower bound and we compare this result with the limiting dyadic diaphony of a random sample.The first author is supported by the Australian Research Council under its Center of Excellence Program.The second author is supported by the Austrian Research Foundation (FWF), Project S 8305 and Project P17022-N12.     

Completely empty pyramids on integer lattices and two-dimensional faces of multidimensional continued fractions

In this paper we develop an integer-affine classification of three-dimensional multistory, completely empty convex marked pyramids. We apply it to obtain the complete lists of compact two-dimensional faces of multidimensional continued fractions lying in planes at integer distances 2, 3, 4, …  to the origin. The faces are considered up to the action of the group of integer-linear transformations.     

Probability measures on [SIN] groups and some related ideals in group algebras

In 1921, Blichfeldt gave an upper bound on the number of integral points contained in a convex body in terms of the volume of the body. More precisely, he showed that , whenever is a convex body containing n + 1 affinely independent integral points. Here we prove an analogous inequality with respect to the surface area F(K), namely . The proof is based on a slight improvement of Blichfeldt’s bound in the case when K is a non-lattice translate of a lattice polytope, i.e., K = t + P, where and P is an n-dimensional polytope with integral vertices. Then we have . Moreover, in the 3-dimensional case we prove a stronger inequality, namely . Authors’ addresses: Martin Henk, Institut für Algebra und Geometrie, Universit?t Magdeburg, Universit?tsplatz 2, D-39106 Magdeburg, Germany; J?rg M. Wills, Mathematisches Institut, Universit?t Siegen, ENC, D-57068 Siegen, Germany     

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     

A Discontinuous Colombeau Differential Calculus

Starting from the Colombeau Generalized Functions, the sharp topologies and the notion of generalized points, we introduce a new kind of differential calculus (for functions between totally disconnected spaces). We also define here the notions of holomorphic generalized functions (in this new framework) and generalized manifold. Finally we give an answer to a question raised in [6].Research partially supported by CNPq (Proc 300652/95-0).     

On a Family of Thue Equations Over Function Fields

Many families of parametrized Thue equations over number fields have been solved recently. In this paper we consider for the first time a family of Thue equations over a polynomial ring. In particular, we calculate all solutions of over for all . The first author was supported by the Austrian Science Foundation, grants S8307-MAT and J2407-N12. The second author was supported by the Austrian Science Foundation, grant S8307-MAT.     

Circularity of Finite Groups without Fixed Points

Let be a fixed point free group given by the presentation where and are relative prime numbers, t = /s and s = gcd( – 1,), and is the order of modulo . We prove that if (1) = 2, and (2) is embeddable into the multiplicative group of some skew field, then is circular. This means that there is some additive group N on which acts fixed point freely, and |((a)+b)((c)+d)| 2 whenever a,b,c,d N, a0c, are such that (a)+b(c)+d.     

A Composition Theorem for Multiple Summing Operators

We prove that the composition S(u₁, …, u_n) of a multilinear multiple 2-summing operator S with 2-summing linear operators u_j is nuclear, generalizing a linear result of Grothendieck. Both authors were partially supported by DGICYT grant BMF2001-1284.     

On a New Continued Fraction Expansion with Non-Decreasing Partial Quotients

We investigate metric properties of the digits occurring in a new continued fraction expansion with non-decreasing partial quotients, the so-called Engel continued fraction (ECF) expansion.