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

[Next] [Prev] [Right] [Left] [Up] [Index] [Root]

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

[Next] [Prev] [Right] [Left] [Up] [Index] [Root]