Small zeros of quadratic forms with linear conditions

Given a quadratic form and M linear forms in N+1 variables with coefficients in a number field K, suppose that there exists a point in K^N+1 at which the quadratic form vanishes and all the linear forms do not. Then we show that there exists a point like this of relatively small height. This generalizes a result of D.W. Masser.     

Bounds for solid angles of lattices of rank three

We find sharp absolute constants C₁ and C₂ with the following property: every well-rounded lattice of rank 3 in a Euclidean space has a minimal basis so that the solid angle spanned by these basis vectors lies in the interval [C₁,C₂]. In fact, we show that these absolute bounds hold for a larger class of lattices than just well-rounded, and the upper bound holds for all. We state a technical condition on the lattice that may prevent it from satisfying the absolute lower bound on the solid angle, in which case we derive a lower bound in terms of the ratios of successive minima of the lattice. We use this result to show that among all spherical triangles on the unit sphere in RN with vertices on the minimal vectors of a lattice, the smallest possible area is achieved by a configuration of minimal vectors of the (normalized) face centered cubic lattice in R³. Such spherical configurations come up in connection with the kissing number problem.     

Integral Points of Small Height Outside of a Hypersurface

Let F be a non-zero polynomial with integer coefficients in N variables of degree M. We prove the existence of an integral point of small height at which F does not vanish. Our basic bound depends on N and M only. We separately investigate the case when F is decomposable into a product of linear forms, and provide a more sophisticated bound. We also relate this problem to a certain extension of Siegel’s Lemma as well as to Faltings’ version of it. Finally we exhibit an application of our results to a discrete version of the Tarski plank problem.     

Frobenius Problem and the Covering Radius of a Lattice

Let and let be relatively prime integers. The Frobenius number of this N-tuple is defined to be the largest positive integer that cannot be expressed as are non-negative integers. The condition that implies that such a number exists. The general problem of determining the Frobenius number given N and is NP-hard, but there have been a number of different bounds on the Frobenius number produced by various authors. We use techniques from the geometry of numbers to produce a new bound, relating the Frobenius number to the covering radius of the null-lattice of this N-tuple. Our bound is particularly interesting in the case when this lattice has equal successive minima, which, as we prove, happens infinitely often.     

Artin t-motifs

We show that analytically trivial t-motifs satisfy a Tannakian duality, without restrictions on the base field, save for that it be of generic characteristic. We show that the group of components of the t-motivic Galois group coincides with the absolute Galois group of the base field.     

Permutation statistics related to a class of noncommutative symmetric functions and generalizations of the Genocchi numbers

We prove conjectures of the third author [L. Tevlin, Proc. FPSAC’07, Tianjin] on two new bases of noncommutative symmetric functions: the transition matrices from the ribbon basis have nonnegative integral coefficients. This is done by means of two composition-valued statistics on permutations and packed words, which generalize the combinatorics of Genocchi numbers.      

The analysis of 1-propanol and 2-propanol in humid air samples using selected ion flow tube mass spectrometry

Following the observation that propanol is present in the breath samples of cystic fibrosis (CF) patients infected by Pseudomonas aeruginosa (PA), a study of the reactions of H(3)O(+), NO(+) and O(2) (+.) with 1-propanol and 2-propanol has been conducted using selected ion flow tube mass spectrometry (SIFT-MS). In this study the number and the distribution of the product ions from NO(+) reactions with the two propanol isomers under humid air conditions were able to differentiate between the two isomers. The reaction mechanisms and the structures of the product ions for these reactions, especially those with H(3)O(+) and NO(+), have been proposed. As an example, 2-propanol was shown to be present in a breath sample from one CF patient infected with PA, and also in a PA isolate from another CF patient grown on Pseudomonas-selective media. The results of this study allow an analytical procedure to be advanced for the analysis of the two propanol isomers, which can also be utilised in other applications.     

On well-rounded sublattices of the hexagonal lattice

We produce an explicit parameterization of well-rounded sublattices of the hexagonal lattice in the plane, splitting them into similarity classes. We use this parameterization to study the number, the greatest minimal norm, and the highest signal-to-noise ratio of well-rounded sublattices of the hexagonal lattice of a fixed index. This investigation parallels earlier work by Bernstein, Sloane, and Wright where similar questions were addressed on the space of all sublattices of the hexagonal lattice. Our restriction is motivated by the importance of well-rounded lattices for discrete optimization problems. Finally, we also discuss the existence of a natural combinatorial structure on the set of similarity classes of well-rounded sublattices of the hexagonal lattice, induced by the action of a certain matrix monoid.     

On the Geometry of Cyclic Lattices

Cyclic lattices are sublattices of \(\mathbb Z^N\) that are preserved under the rotational shift operator. Cyclic lattices were introduced by Micciancio (FOCS, IEEE Computer Society, pp 356–365, 2002) and their properties were studied in the recent years by several authors due to their importance in cryptography. In particular, Peikert and Rosen (Theory of Cryptography, Lecture Notes in Computer Science, vol 3876. Springer, Berlin, pp 145–166, 2006) showed that on cyclic lattices in prime dimensions, the shortest independent vectors problem SIVP reduces to the shortest vector problem SVP with a particularly small loss in approximation factor, as compared to general lattices. In this paper, we further investigate geometric properties of cyclic lattices, proving that a positive proportion of them in every dimension is well-rounded. One implication of our main result is that SVP is equivalent to SIVP on a positive proportion of cyclic lattices in every dimension. As an example, we demonstrate an explicit construction of a family of cyclic lattices on which this equivalence holds. To conclude, we introduce a class of sublattices of \(\mathbb Z^N\) closed under the action of subgroups of the permutation group \(S_N\) , which are a natural generalization of cyclic lattices, and show that our results extend to all such lattices closed under the action of any \(N\) -cycle.