Release Notes V1.20-1 (8 January 1996) since June 1995

Magma V1.20-1 (8 January 1996) since June 1, 1995

Here is a terse summary of the new features installed in Magma between June 1, 1995 and January 8, 1996 (the release date for V1.2). This is intended to show the new features of V1.2 incorporated since June 1995.

Change of Semantics

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

Permutation Groups

• 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, elementary abelian, abelian, nilpotent, soluble, simple, perfect, regular
• Presentation on given generators (for groups of moderate order)

Matrix Groups

• The Murray-O'Brien base selection strategy
• Stabilizer of a subspace
• Soluble quotient
• Composition factors
• Characteristic series: Derived, upper central, lower central, p-central, Jennings
• Characteristic subgroups: Centre, solvable residual
• Subgroup constructions: Centralizer, intersection, core
• Sylow p-subgroup
• Conjugacy classes of elements, class map, power map
• Character table
• 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

Finite Soluble Groups

• Permutation representation on the cosets of a subgroup
• 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

The Rational Field Q and its Ring of Integers Z

• Alternative representation of integers in factored form
• Faster greatest common divisor algorithm for integers
• 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
• Resultant (sub-resultant algorithm, Euclidean algorithm), discriminant
• Factorization of polynomials over large prime finite fields (Shoup algorithm)
• Factorization of polynomials over p-adic fields

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

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

Number Fields and their Orders

• Solution of norm equations
• 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)

• 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)

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

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

• 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