Springer, 2018. — 753 p. — ISBN 3319705652.This book presents state-of-the-art research and survey articles that highlight work done within the Priority Program SPP 1489 “Algorithmic and Experimental Methods in Algebra, Geometry and Number Theory”, which was established and generously supported by the German Research Foundation (DFG) from 2010 to 2016. The goal of the program was to substantially advance algorithmic and experimental methods in the aforementioned disciplines, to combine the different methods where necessary, and to apply them to central questions in theory and practice. Of particular concern was the further development of freely available open source computer algebra systems and their interaction in order to create powerful new computational tools that transcend the boundaries of the individual disciplines involved. The book covers a broad range of topics addressing the design and theoretical foundations, implementation and the successful application of algebraic algorithms in order to solve mathematical research problems. It offers a valuable resource for all researchers, from graduate students through established experts, who are interested in the computational aspects of algebra, geometry, and/or number theory.Algorithmic Aspects of Units in Group Rings A Constructive Approach to the Module of Twisted Global Sections on Relative Projective Spaces Local to Global Algorithms for the Gorenstein Adjoint Ideal of a Curve Picard Curves with Small Conductor Normaliz 2013–2016 Integral Frobenius for Abelian Varieties with Real Multiplication Monodromy of the Multiplicative and the Additive Convolution Constructing Groups of ‘Small’ Order: Recent Results and Open Problems Classifying Nilpotent Associative Algebras: Small Coclass and Finite Fields Desingularization of Arithmetic Surfaces: Algorithmic Aspects Moduli Spaces of Curves in Tropical Varieties Tropical Moduli Spaces of Stable Maps to a Curve Invariant Bilinear Forms on W-Graph Representations and Linear Algebra Over Integral Domains Tropical Computations in polymake Focal Schemes to Families of Secant Spaces to Canonical Curves Inductive and Recursive Freeness of Localizations of Multiarrangements Toric Ext and Tor in polymake and Singular: The Two-Dimensional Case and Beyond The Differential Dimension Polynomial for Characterizable Differential Ideals Factorization of Z -Homogeneous Polynomials in the First q-Weyl Algebra Complexity of Membership Problems of Different Types of Polynomial Ideals Localizations of Inductively Factored Arrangements One Class Genera of Lattice Chains Over Number Fields polyDB: A Database for Polytopes and Related Objects Construction of Neron Desingularization for Two Dimensional Rings A Framework for Computing Zeta Functions of Groups, Algebras, and Modules On Decomposition Numbers of Diagram Algebras Koblitz’s Conjecture for Abelian Varieties Chabauty Without the Mordell-Weil Group An Explicit Theory of Heights for Hyperelliptic Jacobians of Genus Three Some Recent Developments in Spectrahedral Computation Topics on Modular Galois Representations Modulo Prime Powers
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