Ring of integers

About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.

Quantitative spectral gap for thin groups of hyperbolic isometries

Abstract: Let $\Lambda$ be a subgroup of an arithmetic lattice in SO(n+1,1). The quotient $\mathbb{H}^{n+1} / \Lambda$ has a natural family of congruence covers corresponding to primes in some ring of integers. We establish a super-strong approximation result for Zariski-dense $\Lambda$ with some additional regularity and thickness properties. Concretely, this asserts a quantitative spectral gap for the Laplacian operators on the congruence covers. This generalizes results of Sarnak and Xue (1991) and Gamburd (2002).

The characterization of cyclic cubic fields with power integral bases

Abstract: We provide an equivalent condition for the monogenity of the ring of integers of any cyclic cubic field. We show that if a cyclic cubic field is monogenic then it is a simplest cubic field $K_t$ which is the splitting field of a Shanks cubic polynomial $f_t(x):=x^3-tx^2-(t + 3)x-1$ with $t \in \mathbb Z$. Moreover we give an equivalent condition for when $K_t$ is monogenic, which is explicitly written in terms of $t$.

Subalgebras of an algebra with a single generator are finitely generated

Abstract: In order to prove the assertion above it clearly suffices to consider only the free algebra on one generator and hence the polynomials in one variable over a field F. The fact that the sub-algebras of F[x] are all finitely generated is perhaps not surprising, but the author has not been able to discover any proof of it in the literature. (Incidentally, the corresponding statement with the field F replaced by the ring of integers is not true, for the ring of all integral polynomials with even coefficients is clearly not finitely generated.) The proof, though quite simple, would seem to place the problem outside the category of elementary exercises since the solution, at least as presented here, uses some "theory," namely the ever useful facts about modules over principal ideal rings. We shall prove a somewhat stronger result, namely

Lattices that Admit Logarithmic Worst-Case to Average-Case Connection Factors.

Abstract: We demonstrate an average-case problem which is as hard as finding γ(n)-approximate shortest vectors in certain n-dimensional lattices in the worst case, where γ(n) = O( √ log n). The previously best known factor for any class of lattices was γ(n) = O(n). To obtain our results, we focus on families of lattices having special algebraic structure. Specifically, we consider lattices that correspond to ideals in the ring of integers of an algebraic number field. The worst-case assumption we rely on is that in some `p length, it is hard to find approximate shortest vectors in these lattices, under an appropriate form of preprocessing of the number field. Our results build upon prior works by Micciancio (FOCS 2002), Peikert and Rosen (TCC 2006), and Lyubashevsky and Micciancio (ICALP 2006). For the connection factors γ(n) we achieve, the corresponding decisional promise problems on ideal lattices are not known to be NP-hard; in fact, they are in P. However, the search approximation problems still appear to be very hard. Indeed, ideal lattices are well-studied objects in computational number theory, and the best known algorithms for them seem to perform no better than the best known algorithms for general lattices. To obtain the best possible connection factor, we instantiate our constructions with infinite families of number fields having constant root discriminant. Such families are known to exist and are computable, though no efficient construction is yet known. Our work motivates the search for such constructions. Even constructions of number fields having root discriminant up to O(n2/3− ) would yield connection factors better than the current best of O(n). ∗SRI International, cpeikert@alum.mit.edu †Harvard CRCS, DEAS, alon@eecs.harvard.edu

An Integral Digit Derivative Basis for Carlitz Prime Power Torsion Extensions

Abstract: Let $\mathfrak{p}$ be a monic irreducible polynomial in $A:=\mathbb{F}_q[\theta]$, the ring of polynomials in the indeterminate $\theta$ over the finite field $\mathbb{F}_q$, and let $\zeta$ be a root of $\mathfrak{p}$ in an algebraic closure of $\mathbb{F}_q(\theta)$. For each positive integer $n$, let $\lambda_n$ be a generator of the $A$-module of Carlitz $\mathfrak{p}^n$-torsion. We give a basis for the ring of integers $A[\zeta,\lambda_n] \subset K(\zeta, \lambda_n)$ over $A[\zeta] \subset K(\zeta)$ which consists of monomials in the hyperderivatives of the Anderson-Thakur function $\omega$ evaluated at the roots of $\mathfrak{p}$. We also give an explicit field normal basis for these extensions. This builds on (and in some places, simplifies) the work of Angl\`es-Pellarin.