Communications in Mathematics |

Editors: Camilla Hollanti & Lenny Fukshansky

This paper is dedicated to providing an introduction into multidimensionalinteger trigonometry. We start with an exposition of integer trigonometry intwo dimensions, which was introduced in 2008, and use this to generalise theseinteger trigonometric functions to arbitrary dimension. We then move on tostudy the basic properties of integer trigonometric functions. We find integertrigonometric relations for transpose and adjacent simplicial cones, and forthe cones which generate the same simplices. Additionally, we discuss therelationship between integer trigonometry, the Euclidean algorithm, andcontinued fractions. Finally, we use adjacent and transpose cones to introducea notion of best approximations of simplicial cones. In two dimensions, thisnotion of best approximation coincides with the classical notion of the bestapproximations of real numbers.

In this paper we prove that uniform Diophantine exponents of lattices attainonly trivial values.

In this short survey we want to present some of the impact of Minkowski'ssuccessive minima within Convex and Discrete Geometry. Originally related tothe volume of an $o$-symmetric convex body, we point out relations of thesuccessive minima to other functionals, as e.g., the lattice point enumeratoror the intrinsic volumes and we present some old and new conjectures aboutthem. Additionally, we discuss an application of successive minima to a versionof Siegel's lemma.

We prove a fairly general inequality that estimates the number of latticepoints in a ball of positive radius in general position in a Euclidean space.The bound is uniform over lattices induced by a matrix having a boundedoperator norm.

We give an overview of universal quadratic forms and lattices, focusing onthe recent developments over the rings of integers in totally real numberfields. In particular, we discuss indecomposable algebraic integers as one ofthe main tools.

We describe a decisional attack against a version of the PLWE problem inwhich the samples are taken from a certain proper subring of large dimension ofthe cyclotomic ring $\mathbb{F}_q[x]/(\Phi_{p^k}(x))$ with $k>1$ in the casewhere $q\equiv 1\pmod{p}$ but $\Phi_{p^k}(x)$ is not totally split over$\mathbb{F}_q$. Our attack uses the fact that the roots of $\Phi_{p^k}(x)$ oversuitable extensions of $\mathbb{F}_q$ have zero-trace and has overwhelmingsuccess probability as a function of the number of input samples. Animplementation in Maple and some examples of our attack are also provided.

In this paper we give a first study of perfect copositive $n \times n$matrices. They can be used to find rational certificates for completelypositive matrices. We describe similarities and differences to classicalperfect, positive definite matrices. Most of the differences occur only for $n\geq 3$, where we find for instance lower rank and indefinite perfect matrices.Nevertheless, we find for all $n$ that for every classical perfect matrix thereis an arithmetically equivalent one which is also perfect copositive.Furthermore we study the neighborhood graph and polyhedral structure of perfectcopositive matrices. As an application we obtain a new characterization of thecone of completely positive matrices: It is equal to the set of nonnegativematrices having a nonnegative inner product with all perfect copositivematrices.

The integer point transform $\sigma_\PP$ is an important invariant of arational polytope $\PP$, and here we show that it is a complete invariant. Weprove that it is only necessary to evaluate $\sigma_\PP$ at one algebraic pointin order to uniquely determine $\PP$, by employing the Lindemann-Weierstrasstheorem. Similarly, we prove that it is only necessary to evaluate the Fouriertransform of a rational polytope $\PP$ at a single algebraic point, in order touniquely determine $\PP$. We prove that identical uniqueness results also holdfor integer cones. In addition, by relating the integer point transform to finite Fouriertransforms, we show that a finite number of \emph{integer point evaluations} of$\sigma_\PP$ suffice in order to uniquely determine $\PP$. We also give anequivalent condition for central symmetry of a finite point set, in terms ofthe integer point transform, and prove some facts about its local maxima. Mostof the results are proven for arbitrary finite sets of integer points in$\R^d$.

Multilevel lattice codes, such as those associated to Constructions $C$,$\overline{D}$, D and D', have relevant applications in communications. In thispaper, we investigate some properties of lattices obtained via Constructions Dand D' from $q$-ary linear codes. Connections with Construction A, generatormatrices, expressions and bounds for the lattice volume and minimum distancesare derived. Extensions of previous results regarding construction and decodingof binary and $p$-ary linear codes ($p$ prime) are also presented.

In this paper, we find criteria for when cyclic cubic and cyclic quarticfields have well-rounded ideal lattices. We show that every cyclic cubic fieldhas at least one well-rounded ideal. We also prove that there exist families ofcyclic quartic fields which have well-rounded ideals and explicitly constructtheir minimal bases. In addition, for a given prime number $p$, if a cyclicquartic field has a unique prime ideal above $p$, then we provide the necessaryand sufficient conditions for that ideal to be well-rounded. Moreover, incyclic quartic fields, we provide the prime decomposition of all odd primenumbers and construct an explicit integral basis for every prime ideal.

In this editorial survey we introduce the special issue of the journalCommunications in Mathematics on the topic in the title of the article. Ourmain goal is to briefly outline some of the main aspects of this important areaat the intersection of theory and applications, providing the context for thearticles showcased in this special issue.