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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