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MAGMA Computational Algebra System

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Release Notes V1.20-1 (8 January 1996)




Magma V1.20-1 (8 January 1996)


Here is a terse summary of the new features installed
in Magma between November 1, 1994 (the date V1.1 started shipping) and
January 8, 1996 (the release date for V1.2).









Change of Semantics


    

  • 
NormalSubgroups no longer synonym for NormalLattice (will
now return a sequence of subgroup records)







Finitely Generated Abelian Groups (New Category)




    

  • 
Construction as a quotient of a free abelian group 

  • 
Direct product, free product

  • 
Arithmetic 

  • 
Construction of subgroups and quotient groups

  • 
Elementary divisors, primary invariants

  • 
Factor basis, divisor basis, primary basis

  • 
Torsion subgroup, torsion-free subgroup, p-primary component

  • 
Homomorphisms: Image, kernel, cokernel

  • 
Composition series, maximal subgroups, subgroup lattice (for a
finite group)

  • 
Character table (for a finite group)

  • 
Abelian quotient of any group (with its natural homomorphism) 

  • 
Conversion between Z-modules and abelian groups

  • 
Functors from rings and fields onto abelian groups







Permutation Groups




    

  • 
The Holt-Leedham-Green-O'Brien random element algorithm

  • 
Abelian quotient, soluble quotient

  • 
Chief series, chief factors

  • 
Elementary abelian series, radical, solvable residual

  • 
Socle of a primitive group

  • 
An improved algorithm for constructing the O'Nan-Scott decomposition 
of a primitive group

  • 
The performance of the Sylow p-subgroup algorithm has been greatly 
improved

  • 
Normal subgroups

  • 
Conjugacy classes of subgroups, poset of subgroup classes

  • 
Conjugacy classes of subgroups satisfying a condition: Cyclic,
abelian, nilpotent, soluble, simple,
perfect, regular

  • 
Presentation on given generators (for groups of moderate order)

  • 
KG-module corresponding to an elementary abelian section

  • 
Irreducible representations (for groups of moderate order)







Matrix Groups




    

  • 
The Holt-Leedham-Green-O'Brien random element generation algorithm

  • 
The Murray-O'Brien base selection strategy

  • 
Actions on general G-sets

  • 
Stabilizer of a subspace

  • 
Abelian quotient, soluble quotient

  • 
Composition series, composition factors, chief series, chief factors

  • 
Characteristic series: Derived, upper central, lower central,
elementary abelian, p-central, Jennings

  • 
Characteristic subgroups: Centre, derived subgroup, Fitting subgroup,
O_p(G), radical, solvable residual

  • 
Subgroup constructions: Centralizer, intersection,
normal closure, core

  • 
Sylow p-subgroup

  • 
Conjugacy of elements

  • 
Conjugacy classes of elements, class map, power map

  • 
Character table

  • 
Presentation on given generators (for groups of moderate order)

  • 
KG-module corresponding to an elementary abelian section

  • 
Molien series

  • 
Decomposition of a subgroup of GL(n, q) into primitive components 

  • 
Decomposition of a subgroup of GL(n, q) as a tensor product 

  • 
Permutation representation on the cosets of a subgroup

  • 
Conversion of a soluble matrix group to a pc-group







Finite Soluble Groups




    

  • 
Permutation representation on the cosets of a subgroup

  • 
Composition series

  • 
Conjugacy classes of complements of a normal subgroup

  • 
Normal subgroups

  • 
Conjugacy classes of subgroups, poset of subgroup classes

  • 
Conjugacy classes of subgroups satisfying a condition: 
Cyclic, elementary abelian, abelian, nilpotent

  • 
Automorphism group and isomorphism of p-groups (explicit
isomorphism returned)

  • 
New version of Eamonn O'Brien's p-group generation program

  • 
System of double coset representatives for a pair of subgroups

  • 
KG-module corresponding to an elementary abelian section

  • 
Irreducible representations (for groups of moderate order)







The Rational Field Q and its Ring of Integers Z




    

  • 
Alternative representation of integers in factored form

  • 
Faster greatest common divisor algorithm for integers

  • 
More efficient arithmetic for Q

  • 
Fast SQUFOF factorization for small integers

  • 
Multiple polynomial quadratic sieve method of factorization

  • 
Cunningham database of factorizations of integers of
the form p^n +/- 1







Univariate Polynomial Rings




    

  • 
Karatsuba algorithm for polynomial multiplication

  • 
GCD-HEU algorithm for gcd's of polynomials over Z

  • 
Modular algorithm for gcd's of polynomials over number fields

  • 
Resultant (sub-resultant algorithm, Euclidean algorithm),
discriminant

  • 
Factorization of polynomials over large prime finite fields
(Shoup algorithm)

  • 
Factorization of polynomials over Z (Collins-Encarnacion algorithm)

  • 
Factorization of polynomials over number fields (Trager algorithm)

  • 
Factorization of polynomials over p-adic fields

  • 
Construction of ideals and subrings (over K)

  • 
Construction of quotient rings 

  • 
Arithmetic with ideals (over K)

  • 
Determination of whether an ideal is: maximal, prime, primary

  • 
Radical of an ideal (over K)

  • 
Primary decomposition of ideals (over K)

  • 
K[x] / I as an algebra over K







Multivariate Polynomial Rings




    

  • 
Monomial orders: lexicographical, graded lexicographical, graded
      reverse lexicographical, block elimination.

  • 
Arithmetic with elements

  • 
Recursive coefficient, monomial, term, and degree access

  • 
Determination of whether an element is: a unit, a zero-divisor,
nilpotent

  • 
Differentiation, integration, evaluation and interpolation

  • 
Resultant (sub-resultant algorithm), discriminant

  • 
Greatest common divisor (sparse EEZ-GCD and fast GCD-HEU algorithms)

  • 
Factorization over Z, Q, and rational function fields over Z, Q
(EEZ algorithm).

  • 
Construction of ideals and subrings

  • 
Construction of quotient rings 

  • 
Groebner bases of ideals

  • 
Groebner Walk algorithm for converting the Groebner basis of an
ideal with respect to one order to the Groebner basis of the ideal
with respect to another order

  • 
Arithmetic with ideals (sum, product, intersection, colon ideal,
leading monomial ideal)

  • 
Normal form of a polynomial with respect to an ideal

  • 
S-polynomial of two polynomials

  • 
Determination of whether a polynomial is in an ideal or its radical

  • 
Determination of whether an ideal is: zero, proper, zero-dimensional

  • 
Computation of the variety of a zero-dimensional ideal

  • 
Elimination ideals

  • 
Relation ideals (determination of algebraic relations between
polynomials)

  • 
Syzygy modules

  • 
Construction of elementary symmetric polynomials

  • 
Determination of whether a polynomial is symmetric, and,
if so, expression of it in terms of the elementary symmetric
polynomials

  • 
Permutation and matrix group actions on polynomials







Valuation Rings




    

  • 
Construction of a valuation ring of a rational function field

  • 
Arithmetic







Quadratic Fields and their Orders




    

  • 
Discriminant, signature, conductor

  • 
Maximal order, integral basis 

  • 
Decomposition of primes

  • 
Quadratic forms: Reduction, composition, powering, prime form,
action of SL(2,Z)

  • 
Class number (Shank's algorithm)

  • 
Ideal class group (Buchmann's method)

  • 
Unit group, fundamental unit, regulator 

  • 
Factorization of an ideal

  • 
Solution of norm equations







General Number Fields and their Orders




    

  • 
Relative extensions

  • 
Kummer extensions

  • 
Improved ideal arithmetic

  • 
Determination of whether an ideal is: integral, prime, principal

  • 
Decomposition of primes 

  • 
Improved algorithms for class group unit group

  • 
Solution of relative norm equations and Thue equations

  • 
Determination of subfields

  • 
Isomorphism of number fields

  • 
Determination of Galois groups (for equations of degree less
than 12)







The Real and Complex Fields (PARI real package)




    

  • 
Arithmetic

  • 
Square root, arithmetic-geometric mean

  • 
Continued fraction expansion of a real number

  • 
Constants: pi, Euler's constant, Catalan's constant 

  • 
Logarithm, dilogarithm, exponential

  • 
Trigonometric functions and their inverses

  • 
Hyperbolic functions and their inverses

  • 
Bernoulli numbers

  • 
Gamma function, incomplete Gamma function, complementary incomplete
Gamma function, logarithm of Gamma function

  • 
J-Bessel function, K-Bessel function

  • 
U-confluent hypergeometric function

  • 
Logarithmic integral, exponential integral 

  • 
Error function, complementary error function

  • 
Dedekind eta function

  • 
Jacobi sine theta-function and its k-th derivative

  • 
Log derivative (psi) function, i.e, {Gamma'(x)over Gamma(x)}

  • 
Riemann-zeta function

  • 
Polylogarithm, Zagier's modifications of the polylogarithm

  • 
Weber's f-function, Weber's f_2-function

  • 
Integer polynomial having a given real or complex number as
an approximate root (Hastad, Lagarias and Schnorr LLL-method)

  • 
Roots of a polynomial (Schonhage method)

  • 
Summation of a series (Euler-Wijngaarden method
for alternating series)

  • 
Numerical integration of a function (Romberg-type methods)







p-adic and Local Fields (New Categories)




    

  • 
Arithmetic operations; valuation of an element

  • 
Square root, n-root, logarithm, exponential

  • 
Ring of integers of a p-adic field

  • 
Construction of an unramified extension of a p-adic field

  • 
Construction of a totally ramified extension

  • 
Factorization of polynomials over a p-adic field







Modules over multivariate polynomial rings




    

  • 
Arithmetic

  • 
Row and column operations on elements

  • 
Membership test

  • 
Construction of submodules

  • 
Sum and intersection of submodules

  • 
Image, kernel

  • 
Syzygy modules







Matrix Algebras




    

  • 
Quotient algebras

  • 
Ideals

  • 
Sum, intersection, product of ideals

  • 
Centre, commutator algebra, Jacobson radical

  • 
Maximal (minimal) left, right, two-sided ideals







Enumerative Combinatorics




    

  • 
Factorial

  • 
Binomial, multinomial coefficients

  • 
Number of partitions of n

  • 
Enumeration of restricted and unrestricted partitions

  • 
Stirling numbers of the first and second kind

  • 
Fibonacci numbers







Graphs




    

  • 
Characteristic polynomial, spectrum







Incidence Structures and Designs (New Category)




    

  • 
Parameters, intersection numbers for an incidence structure

  • 
Unary operations: complement, contraction, dual, residual

  • 
Binary operations: sum, union

  • 
Development of a design from a block

  • 
Standard symmetric difference sets

  • 
Starter blocks for general designs: standard examples

  • 
Properties: balanced, complete, self-dual, simple

  • 
Graphs from designs: block graph, incidence graph, point graph

  • 
Automorphism group (J. Leon's algorithm), canonical labelling

  • 
Testing pairs of designs for isomorphism

  • 
Group actions on a design: orbits and stabilizers of points
and blocks 

  • 
Symmetry properties: point transitive, block transitive







Error-correcting Codes




    

  • 
Trace of a code

  • 
Code words of a designated weight

  • 
Speed-up of determination of weight distribution/enumerator

  • 
Complete weight enumerator

  • 
MacWilliams transform for complete weight enumerator

  • 
Group actions on a code







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Mon Nov 16 15:04:45 EST 2009



        


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