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Collaborators

The Magma system has benefited enormously from contributions made by many members of the mathematical community. We list below those persons and research groups who have given the project substantial assistance either by allowing us to adapt their software for inclusion within Magma or through general advice and criticism. We wish to express our gratitude both to the people listed here and to all those others who participated in some aspect of the Magma development.

Richard Brent

Henri Cohen

George Collins

John Cremona

Greg Gamble

Xavier Gourdon

Willem de Graaf

Markus Grassl

Charles Leedham-Green

George Havas

Derek Holt

Alexander Hulpke

Gregor Kemper

Arjen Lenstra

Jeff Leon

Jin Xian Lian

Steve Linton

Rüdiger Loos

Brendan McKay

Torsten Minkwitz

Eamonn O'Brien

Francois Morain

Michel Olivier

Michael Pohst

Gerhard Schneider

Charlie Sims

Bernd Souvignier

Henry Spencer

William Stein

• A facility for computing with arbitrary but fixed precision reals was based on Richard Brent's (ANU) FORTRAN package MP. Richard also provided his database of 180, 000 factorizations of integers of the form p^(n +- 1), together with his intelligent factorization code FACTOR.

• The group headed by Henri Cohen (Bordeaux) made available their excellent Pari system for computational number theory for inclusion in Magma. Pascal Letard of the Pari group visted Sydney for two months in 1994 and recoded large sections of Pari for Magma. The Pari facilities installed in Magma include arithmetic for real and complex fields (the `free' model), approximately 100 special functions for real and complex numbers, power series, p-adic numbers, factorization of polynomials over p-adic fields, binary quadratic forms, quadratic fields and other features.

• The package for multivariate polynomials in early versions of Magma was based on the SAC-2 package of George Collins and Rüdiger Loos. The current Magma function for the factorization of univariate polynomials over the integers owes much to the help and advice of George Collins.

• The Magma facility for elliptic curves over the rational field is heavily based on John Cremona's mwrank package.

• Greg Gamble (UWA) helped refine the concept of a G-set for a permutation group and drafted several sections of the chapter on permutation groups.

• Xavier Gourdon (INRIA, Paris) made available his C implementation of A. Schönhage's splitting-circle algorithm for the fast computation of the roots of a polynomial to a specified precision. Xavier also assisted with the adaption of his code for the Magma kernel.

• The major structural machinery for Lie algebras has been implemented for Magma by Willem de Graaf and is based on his ELIAS package originally written in GAP.

• Markus Grassl (IAKS, Karlsruhe) has contributed the ConcatenatedCode and JustensenCode constructions, as well as many other suggestions in Coding Theory.

• Charles Leedham-Green (QMW) has worked with the Magma group on the design of a large number of algorithms for soluble groups and local fields during several visits to Sydney.

• The Todd-Coxeter procedure in Magma was designed by George Havas (University of Queensland) and implemented by Jin Xian Lian (also of the University of Queensland). The Magma Smith and Hermite Normal Form code together with the underlying extended gcd algorithms were developed in close cooperation with George Havas and Bohdan Majewski (University of Newcastle). George has also contributed to the design of the machinery for finitely presented groups.

• Cohomology calculations for permutation groups are implemented by a program written by Derek Holt. His program, kbmag, for monoids and automatic groups has also been installed in Magma. Derek has contributed to the design of algorithms in module theory and permutation groups.

• Alexander Hulpke has made available his database of all transitive permutation groups of degree up to 22. This incorporates the earlier database of Greg Butler and John McKay containing all transitive groups of degree up to 15.

• Gregor Kemper (Heidelberg) has contributed all of the major algorithms of the Invariant Theory module of Magma, together with many other helpful suggestions in the area of Commutative Algebra.

• The main integer factorization tools available in Magma are due to A.K. Lenstra (Bellcore) and his collaborators. These include the elliptic curve method and a multiple polynomial quadratic sieve developed by Arjen from his `factoring by email' MPQS during a visit to Sydney in 1995.

• The PERM package developed by Jeff Leon (UIC) for efficient backtrack searching in permutation groups is used for most of the permutation group constructions that employ backtrack search. Jeff's programs for finding automorphism groups of codes, designs and matrices are also used.

• The vector enumeration program of Steve Linton (St. Andrews) provides Magma with the capability of constructing matrix representations for finitely presented associative algebras.

• Automorphism groups and isomorphism testing of graphs is performed by Brendan McKay's program nauty.

• An implementation of the Dixon-Minkwitz algorithm for computing the ordinary irreducible representations of a finite group was developed with Torsten Minkwitz's assistance.

• The p-quotient program, developed by Eamonn O'Brien based on earlier work by George Havas and Mike Newman, provides a key facility for studying p-groups in Magma. Eamonn's extensions of this package for generating p-groups and computing automorphism groups of p-groups are also included in Magma. The corresponding sections of the Handbook were written by Eamonn.

• The primality of integers is proven using the ECPP (Elliptic Curves and Primality Proving) package written by Francois Morain (Ecole Polytechnique and INRIA). The ECPP program in turn uses the BigNum package developed jointly by INRIA and Digital PRL.

• A program developed by Michel Olivier (Bordeaux) is used to compute Galois groups for polynomials having degree less than 12.

• The facilities for general number fields in Magma are provided by the KANT V4 package developed by Michael Pohst and collaborators, originally at Düsseldorf and now at TU, Berlin. This package provides extensive machinery for computing with maximal orders of number fields and their ideals. Particularly noteworthy are functions for computing the class group, the unit group, systems of fundamental units, and subfields of a number field. Johannes Graf von Schmettow, Max Jüntgen and Mario Daberkow, all from the KANT group, have spent extended periods in Sydney working with the Magma group.

• The Magma implementation of the Dixon-Schneider algorithm for computing the table of ordinary characters of a finite group is based on an earlier version written for Cayley by Gerhard Schneider (Karlsruhe).

• The low index subgroup function is implemented by code that is based on a Pascal program written by Charlie Sims (Rutgers).

• The functions for computing automorphism groups and isometries of lattices are based on the AUTO and ISOM programs of Bernd Souvignier.

• The code to perform the regular expression matching in the regexp intrinsic function comes from the V8 regexp package by Henry Spencer (University of Toronto).

• The facilities for computing with modular symbols in Magma were designed and implemented by William A. Stein (Harvard University), with input from David Kohel and Kevin Buzzard.

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