Algorithms in algebraic number theory

In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to be done in the area. We hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them. The discussion is concentrated of three topics: the determination of Galois groups, the determination of the ring of integers of an algebraic number field, and the computation of the group of units and the class group of that ring of integers.Comment: 34 page

Testing isomorphism of lattices over CM-orders

A CM-order is a reduced order equipped with an involution that mimics complex conjugation. The Witt-Picard group of such an order is a certain group of ideal classes that is closely related to the "minus part" of the class group. We present a deterministic polynomial-time algorithm for the following problem, which may be viewed as a special case of the principal ideal testing problem: given a CM-order, decide whether two given elements of its Witt-Picard group are equal. In order to prevent coefficient blow-up, the algorithm operates with lattices rather than with ideals. An important ingredient is a technique introduced by Gentry and Szydlo in a cryptographic context. Our application of it to lattices over CM-orders hinges upon a novel existence theorem for auxiliary ideals, which we deduce from a result of Konyagin and Pomerance in elementary number theory.Comment: To appear in SIAM Journal on Computin

Efficiently Computing Real Roots of Sparse Polynomials

We propose an efficient algorithm to compute the real roots of a sparse polynomial $f\in\mathbb{R}[x]$ having $k$ non-zero real-valued coefficients. It is assumed that arbitrarily good approximations of the non-zero coefficients are given by means of a coefficient oracle. For a given positive integer $L$, our algorithm returns disjoint disks $\Delta_{1},\ldots,\Delta_{s}\subset\mathbb{C}$, with $s<2k$, centered at the real axis and of radius less than $2^{-L}$ together with positive integers $\mu_{1},\ldots,\mu_{s}$ such that each disk $\Delta_{i}$ contains exactly $\mu_{i}$ roots of $f$ counted with multiplicity. In addition, it is ensured that each real root of $f$ is contained in one of the disks. If $f$ has only simple real roots, our algorithm can also be used to isolate all real roots. The bit complexity of our algorithm is polynomial in $k$ and $\log n$, and near-linear in $L$ and $\tau$, where $2^{-\tau}$ and $2^{\tau}$ constitute lower and upper bounds on the absolute values of the non-zero coefficients of $f$, and $n$ is the degree of $f$. For root isolation, the bit complexity is polynomial in $k$ and $\log n$, and near-linear in $\tau$ and $\log\sigma^{-1}$, where $\sigma$ denotes the separation of the real roots

Structure and Phase Transitions of Alkyl Chains on Mica

We use molecular dynamics as a tool to understand the structure and phase transitions [Osman et. al. J. Phys. Chem. B 2000, 104, 4433; 2002, 106, 653] in alkylammonium micas. The consistent force field 91 is extended for accurate simulation of mica and related minerals. We investigate mica sheets with 12 octadecyltrimethylammonium (C18) ions or 12 dioctadecyldimethylammonium (2C18) ions, respectively, as single and layered structures at different temperatures with periodicity in the xy plane by NVT dynamics. The alkylammonium ions reside preferably above the cavities in the mica surface with an aluminum-rich boundary. The nitrogen atoms are 380 to 390 pm distant to the superficial silicon-aluminum plane. With increasing temperature, rearrangements of C18 ions on the mica surface are found, while 2C18 ions remain tethered due to geometric restraints. We present basal-plane spacings in the duplicate structures, tilt angles of the alkyl chains, and gauche-trans ratios to analyze the chain conformation. Also, the individual phase transitions of the two systems on heating are explained. Where experimental data are available, the agreement is very good. We propose a geometric parameter lamba for the saturation of the surface with alkyl chains, which determines the preferred self-assembly pattern, i.e., islands, intermediate, or continuous. Lambda also determines the tilt angles in continuous layers on mica or other surfaces. The thermal decomposition appears to be a Hofmann elimination with mica as a base-template.Comment: 45 pages with 6 tables and 5 figure

Galois module structure of oriented Arakelov class groups

We show that Chinburg's Omega(3) conjecture implies tight restrictions on the Galois module structure of oriented Arakelov class groups of number fields. We apply our findings to formulating a probabilistic model for Arakelov class groups in families, offering a correction of the Cohen--Lenstra--Martinet heuristics on ideal class groups.Comment: 14 pages; comments welcom

Theory of ice premelting in porous media

Premelting describes the confluence of phenomena that are responsible for the stable existence of the liquid phase of matter in the solid region of its bulk phase diagram. Here we develop a theoretical description of the premelting of water ice contained in a porous matrix, made of a material with a melting temperature substantially larger than ice itself, to predict the amount of liquid water in the matrix at temperatures below its bulk freezing point. Our theory combines the interfacial premelting of ice in contact with the matrix, grain boundary melting in the ice, and impurity and curvature induced premelting, the latter occurring in regions which force the ice-liquid interface into a high curvature configuration. These regions are typically found at points where the matrix surface is concave, along contact lines of a grain boundary with the matrix, and in liquid veins. Both interfacial premelting and curvature induced premelting depend on the concentration of impurities in the liquid, which, due to the small segregation coefficient of impurities in ice are treated as homogeneously distributed in the premelted liquid. Our principal result is an equation for the fraction of liquid in the porous medium as a function of the undercooling, which embodies the combined effects of interfacial premelting, curvature induced premelting, and impurities. The result is analyzed in detail and applied to a range of experimentally relevant settings.Comment: 14 pages, 10 figures, accepted for publication in Physical Review

Revisiting the Sectoral Linder Hypothesis: Aggregation Bias or Fixed Costs?

This paper reassesses and revisits the Sectoral Linder Hypothesis due to Hallak (2010), according to which similar tastes for quality lead to more intensive trade between similar countries. First, it will be shown that allowing for strictly non-homothetic preferences reduces confoundedness and improves results. Moreover, the country/firm level extensive margin is taken into account. This approach allows controlling for unobserved firm level heterogeneity and selection bias (Helpman et al. 2008). The advantage in terms of interpretation is that differences in coefficients at the two margins can be linked to fixed cost effects. The attempt is to show that the Linder effect is confounded with fixed (opportunity) costs of trade thereby leading to downward biased results. There is some evidence that this effect is exacerbated at the aggregate, intersectoral level. Fixed (opportunity) costs seem to be higher in sectors where similar countries trade a lot. The evidence reinforces the sectoral Linder hypothesis, and suggests that the patterns might prevail at the more aggregate levels. Other robustness checks suggest that results are not confined to products that are vertically differentiated