Development roadmap for AbstractAlgebra.jl

[wbhart]

AbstractAlgebra.jl intends to provide a generic implementation of finitely presented groups, rings, fields, modules, ideals, in Julia with no C dependencies.

Here is an approximate roadmap for future development:

Basic rings

• Julia integers
• Julia rationals
• Julia floats
• Prime fields GF(p)
• Finite fields (Conway polynomials)
• Number fields (arithmetic only)
• Padic field

Fraction fields

• Basic arithmetic

Residue rings R/(Rf)

• Basic arithmetic

Univariate polynomials

• Basic arithmetic
• Gcd
• Extended gcd
• Resultant (Ducos)
• Evaluation
• Composition
• Substitution
• Interpolation (over integral domain)
• Signature
• Half GCD
• Multimodular GCD over Euclidean domain
• Resultant over non-integral domain

Laurent polynomials

• Basic arithmetic

Relative power series

• Basic arithmetic
• Composition
• Reversion

Laurent series

• Basic arithmetic
• Division over a ring
• Composition
• Reversion

Puiseux series

• Basic arithmetic
• Composition
• Reversion

Lazy power series

• Basic arithmetic
• Composition
• Reversion

Multivariate polynomials

• Reverse order of terms and exponents to match Flint
• Basic arithmetic
• Gcd (psr)
• Resultant (Ducos)
• Evaluation at all variables
• [] Evaluation at one variable
• Composition
• Substitution

Multivariate power series

• Basic arithmetic

Multivariate Laurent series

• Basic arithmetic

Multivariate Puiseux series

• Basic arithmetic

Matrices (dense)

• Basic arithmetic
• Solving
• Inverse
• Determinant
• Rank
• Reduced row echelon form
• LU decomposition
• QR decomposition
• Gram matrix
• Hessenberg form
• Hermite normal form
• Smith normal form
• Popov form
• Frobenius form
• Jordan canonical form
• Rational canonical form
• Kronecker product
• Characteristic polynomial
• Minimal polynomial

Matrices (sparse)

• Basic arithmetic
• Solving
• Inverse
• Determinant
• Rank
• Reduced row echelon form
• LU decomposition
• QR decomposition
• Gram matrix
• Hessenberg form
• Hermite normal form
• Smith normal form
• Popov form
• Frobenius form
• Jordan canonical form
• Rational canonical form
• Kronecker product
• Characteristic polynomial
• Minimal polynomial

Groups

• Permutation groups
• Characters of S_n
• Young tableaux

Function fields

• Rational function fields
• Finite extensions

Maps and morphisms

• Composition
• Inverse
• Identity
• Functional maps
• Linear morphisms
• Partial maps

Modules

• Free modules
• Direct products

Finitely presented modules over euclidean domains

• Element arithmetic
• Submodules
• Quotient modules
• Intersection
• Homomorphisms
• kernels
• Images
• Module structural decomposition

Modules over Dedekind domains

• Element arithmetic
• Homomorphisms

Ideals

• Element arithmetic
• Quotient rings?
• Colon module [M:I]

Polynomials over non-commutative ring

• Basic arithmetic

Matrix algebras

• Basic arithmetic