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This is a systematic account of the multiplicative structure of integers, from the probabilistic point of view. The authors are especially concerned with the distribution of the divisors, which is as fundamental and important as the additive structure of the integers, and yet until now has hardly been discussed outside of the research literature. Hardy and Ramanujan initiated this area of research and it was developed by Erdös in the thirties. His work led to some deep and basic conjectures of wide application which have now essentially been settled. This book contains detailed proofs, some of which have never appeared in print before, of those conjectures that are concerned with the propinquity of divisors. Consequently it will be essential reading for all researchers in analytic number theory.

Divisor theory. --- Distribution (Probability theory)

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Defining and computing a greatest common divisor of two polynomials with inexact coefficients is a classical problem in symbolic-numeric computation. The first part of this book reviews the main results that have been proposed so far in the literature. As usual with polynomial computations, the polynomial GCD problem can be expressed in matrix form: the second part of the book focuses on this point of view and analyses the structure of the relevant matrices, such as Toeplitz, Toepliz-block and displacement structures. New algorithms for the computation of approximate polynomial GCD are presented, along with extensive numerical tests. The use of matrix structure allows, in particular, to lower the asymptotic computational cost from cubic to quadratic order with respect to polynomial degree. .

Algebra. --- Divisor theory. --- Mathematics. --- Polynomials. --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Approximation algorithms. --- Heuristic algorithms --- Mathematical analysis

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This is the first book to deal with invariant theory and the representations of finite groups. By restricting attention to finite groups Dr Benson is able to avoid recourse to the technical machinery of algebraic groups, and he develops the necessary results from commutative algebra as he proceeds. Thus the book should be accessible to graduate students. In detail, the book contains an account of invariant theory for the action of a finite group on the ring of polynomial functions on a linear representation, both in characteristic zero and characteristic p. Special attention is paid to the role played by pseudoreflections, which arise because they correspond to the divisors in the polynomial ring which ramify over the invariants. Also included is a new proof by Crawley-Boevey and the author of the Carlisle-Kropholler conjecture. This volume will appeal to all algebraists, but especially those working in representation theory, group theory, and commutative or homological algebra.

Invariants. --- Finite groups. --- Divisor theory. --- Categories (Mathematics) --- Ideals (Algebra) --- Modules (Algebra) --- Groups, Finite --- Group theory --- Groups (Mathematics)

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This book introduces the theory of complex surfaces through a comprehensive look at finite covers of the projective plane branched along line arrangements. Paula Tretkoff emphasizes those finite covers that are free "ients of the complex two-dimensional ball. Tretkoff also includes background on the classical Gauss hypergeometric function of one variable, and a chapter on the Appell two-variable F1 hypergeometric function.The material in this book began as a set of lecture notes, taken by Tretkoff, of a course given by Friedrich Hirzebruch at ETH Zürich in 1996. The lecture notes were then considerably expanded by Hirzebruch and Tretkoff over a number of years. In this book, Tretkoff has expanded those notes even further, still stressing examples offered by finite covers of line arrangements. The book is largely self-contained and foundational material is introduced and explained as needed, but not treated in full detail. References to omitted material are provided for interested readers.Aimed at graduate students and researchers, this is an accessible account of a highly informative area of complex geometry.

Curves, Elliptic. --- Geometry, Algebraic. --- Projective planes. --- Unit ball. --- Riemann surfaces. --- Surfaces, Riemann --- Functions --- Ball, Unit --- Holomorphic functions --- Planes, Projective --- Geometry, Projective --- Algebraic geometry --- Geometry --- Elliptic curves --- Curves, Algebraic --- Appell hypergeometric function. --- Chern numbers. --- Euler number. --- Friedrich Hirzebruch. --- Gauss hypergeometric function. --- Gaussian curvature. --- Hermitian metric. --- Kodaira dimension. --- Mbius transformation. --- Miyaoka-Yau inequality. --- Riemann surface. --- Riemannian metric. --- algebraic geometry. --- algebraic surface. --- arithmetic monodromy group. --- b-space. --- ball "ient. --- canonical divisor class. --- complete quadrilateral. --- complex 2-ball. --- complex manifold. --- complex surface. --- covering group. --- covering space. --- differential geometry. --- divisor class group. --- divisor. --- elliptic curve. --- finite covering. --- first Chern class. --- fractional linear transformation. --- free 2-ball "ient. --- fundamental group. --- geometry. --- intersection point. --- line arrangement. --- line bundle. --- linear arrangement. --- log-canonical divisor. --- minimal surface. --- monodromy group. --- orbifold structure. --- orbifold. --- partial differential equation. --- plurigenus. --- projective plane. --- proportionality deviation. --- ramification indices. --- rational curve. --- regular point. --- signature. --- solution space. --- topological invariant. --- transversely intersecting divisor. --- triangle groups. --- weight.

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Divisor theory in module categories

Commutative rings. --- Categories (Mathematics) --- Modules (Algebra) --- Divisor theory. --- Ideals (Algebra) --- Finite number systems --- Modular systems (Algebra) --- Algebra --- Finite groups --- Rings (Algebra) --- Category theory (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Topology --- Functor theory