CommutativeRings.jl

W.I.P

This software is the start of a computer algebra system specialized to discrete calculations in the area of integer numbers ℤ, modular arithmetic ℤ/m fractional ℚ, polynomials ℤ[x]. Also multivariate polynomials ℤ[x,y,...] and Galois fields GF(p^r) are supported. The polynomials may be extended to factor rings like ℤ[:x] / (x^2 + 1).

This package is not seen as a replacement of Nemo.jl or AbstractAlgebra.jl, which should be used for serious work. It is understood more like a sandbox to try out a simpler API.

It is important, that rings may be freely combined, for example (ℤ/p)[x] (polynomials over the factor ring for a prime number p), Frac(ℤ[x]), the rational functions with integer coefficients, or GF(64)[:x], polynomials over the Galois field. The factor rings include Ideals, which are of major importance with multivariate polynomials.

The mentioned examples are elementary examples for ring structures. The can be liberately combined to fractional fields, factor rings, and polynomials of previously defined structures.

So it is possible to work with rational functions, which are fractions of polynomials, where the polynomial coefficients are in ℤ/p, for example. In the current standard library we have modules Rational and Polynomial besides the numeric subtypes of Number and some support for modular calculations with integers.

The original motivation for writing this piece of sofware, when I tried to handle polynomials over a factor ring. There was no obvious way of embedding my ring elements into the Julia language and for example exploit polynomial calculations from the Polynomial package for that. There seems to be a correspondence between Julia types and structures and the algebraic stuctures I want to work with. So the idea was born to define abstract and concrete types in Julia, the objects of those types representing the ring elements to operate on. As types are first class objects in Julia it was also possible to define combinations in a language affine way. Also ring homomorphisms, i.e. strucure-respecting mappings between rings (of differnt kind) find a natural representation as one-argument-functions or methods with corresponding domains. The typical canonical homomorphisms, can be conveniently implemented as constructors.

The exploitation of the julia structures is in contrast to the alternative package AbstractAlgebra, which defines separate types for ring elements and the ring classes themselves, the elements keeping an explicit link to the owner structure.

To distinct variants of rings, we use type parameters, for example the m in ℤ/m or the x in ℤ[:x]. Other type parameters may be used to specify implementation restictions, for example typically the integer types used for the representation of the objects.

Correspondence between algebraic and Julia categories:

julia> using CommutativeRings

 # starting with some calculation in the quotient field Z/31

julia> m = 31
31
julia> ZZp = ZZ/m
ZZmod{31,Int8}
julia> modulus(ZZp)
31
julia> z1 = ZZp(12)
12°
julia> z2 = ZZp(17)
17°
julia> z1 + z2
29°
julia> z1 * z2
18°
julia> inv(z1)
13°
julia> 13z1
1°

 # using a big prime number as parameter, the class is identified by an arbitrary symbol (:p)

julia> ZZbig = ZZ / (big(2)^521 - 1)
ZZmod{Symbol("686479766013060971498190079908139321726943530014330540939446345918554318339765
              6052122559640661454554977296311391480858037121987999716643812574028291115057151"), BigInt}

julia> modulus(ZZbig)
686479766013060971498190079908139321726943530014330540939446345918554318339765
6052122559640661454554977296311391480858037121987999716643812574028291115057151
julia> zb = ZZbig(10)
10°
julia> zb^(modulus(ZZbig)-1) # Fermat's little theorem for primes
1°


... # now polynomials with element type of ZZ/31

julia> P = ZZp[:x]
UnivariatePolynomial{ZZmod{31, Int8}, :x}
julia> x = monom(P)
x

julia> p = (x + 2)^2 * (x - 1)
x^3 + 3°*x^2 + 27°
julia> p.coeff
4-element Array{ZZmod{31,Int8},1}:
 -4°
 0°
 3°
 1°
julia> 1 + p
x^3 + 3°*x^2 + 28°


julia> gcd(p, x-1)
x + 30°
julia> p / (x-1)
x^2 + 4°⋅x + 4°
julia> p / (x+2)
x^2 + x + 29°


julia> p / (x + 3) * (x + 1)
ERROR: DomainError with (x^3 + 3°*x^2 + 27°, x + 3°):
cannot divide a / b
Stacktrace:

    # press "]"
    (@v1.8) pkg> add CommutativeRings
    Updating registry at `~/.julia/registries/General.toml`
   Resolving package versions...
  No Changes to `~/.julia/environments/v1.8/Project.toml`
  No Changes to `~/.julia/environments/v1.8/Manifest.toml`
     # press backspace

julia> using CommutativeRings
[ Info: Precompiling CommutativeRings [a6d4fa9c-9e0b-4795-89f3-f481b7b5e384]

(CommutativeRings) pkg> test
Precompiling project...
  1 dependency successfully precompiled in 3 seconds. 15 already precompiled.
     Testing Running tests...
Test Summary: | Pass  Total  Time
typevars      |    8      8  6.9s
Test Summary: | Pass  Total   Time
generic       |   80     80  26.1s
Test Summary: | Pass  Total  Time
ZZ            |  129    129  4.1s
Test Summary: | Pass  Total  Time
QQ            |   49     49  0.7s
Test Summary: | Pass  Total  Time
ZZmod         |  280    280  5.3s
Test Summary: | Pass  Total     Time
galoisfields  |  600    600  1m34.9s
Test Summary: | Pass  Total   Time
fraction      |   50     50  12.5s
Test Summary: | Pass  Total  Time
quotient      |   32     32  1.3s
Test Summary: | Pass  Total  Time
univarpoly    |  302    302  9.9s
Test Summary: | Pass  Total  Time
determinant   |   12     12  2.4s
Test Summary: | Pass  Total  Time
resultant     |   52     52  0.9s
Test Summary: | Pass  Total   Time
multivarpoly  |  174    174  35.7s
Test Summary: | Pass  Total  Time
ideal         |   42     42  9.2s
Test Summary: | Pass  Total  Time
factors       |   33     33  3.7s
Test Summary: | Pass  Total  Time
numbertheory  |   45     45  2.4s
Test Summary: | Pass  Total  Time
enumerations  |  115    115  8.4s
Test Summary: | Pass  Total   Time
linearalgebra |   51     51  36.2s
Test Summary:  | Pass  Total   Time
rationalnormal |   81     81  25.0s
┌ Warning: irreducibility of 2*x^111 + 7*x^74 + x^37 + 1 cannot be proved - p = 5 B = 16323696413041099
└ @ CommutativeRings ~/dev/CommutativeRings/src/intfactorization.jl:243
┌ Warning: irreducibility of 2*x^111 + 7*x^74 + x^37 + 1 cannot be proved - p = 25 B = 16323696413041099
└ @ CommutativeRings ~/dev/CommutativeRings/src/intfactorization.jl:243
┌ Warning: irreducibility of 2*x^111 + 7*x^74 + x^37 + 1 cannot be proved - p = 625 B = 16323696413041099
└ @ CommutativeRings ~/dev/CommutativeRings/src/intfactorization.jl:243
┌ Warning: irreducibility of 2*x^111 + 7*x^74 + x^37 + 1 cannot be proved - p = 390625 B = 16323696413041099
└ @ CommutativeRings ~/dev/CommutativeRings/src/intfactorization.jl:243
┌ Warning: irreducibility of 2*x^111 + 7*x^74 + x^37 + 1 cannot be proved - p = 152587890625 B = 16323696413041099
└ @ CommutativeRings ~/dev/CommutativeRings/src/intfactorization.jl:243
[ Info: irreducibility of 2*x^111 + 7*x^74 + x^37 + 1 proved - p = 23283064365386962890625 B = 16323696413041099
Test Summary: | Pass  Total     Time
intfactors    |   67     67  1m43.2s
Test Summary: | Pass  Total  Time
powerseries   |   82     82  2.3s
Test Summary: | Pass  Total   Time
specialseries |   44     44  15.3s
     Testing CommutativeRings tests passed  

Each complete Julia type (with all type parameters specified) defines a singleton algebraic class.

    p = Int128(2)^127 - 1
    Zp = ZZmod{p,Int128}

For general factor rings and for polynomials there are convenient constructors, which look like the typical mathematical notation R[:x,:y,...] and R / I. Here the symbols :x, :y define the name of the variables of a uni- or multivariate polynomial ring over R. I is an ideal of R or an element of R, which represents the corresponding principal ideal.

    S = ZZ{Int}
    P = S[:x]
    x = monom(P, 1) # same as P([0,1])
    Q = P / (x^2 + 1)

The / notion is not implemented for Julia integer types (would be type piracy) so this doesn't work:

    julia> Int / 31
ERROR: MethodError: no method matching /(::Type{Int64}, ::Int64)

Ideals are can be denoted as Ideal([a, ...]) where a, ... are elements of R or Ideal(a). For convenience, principal ideals support also a * R == Ideal(a) and p*R == Ideal(R(p)), where p is an integer.

    Z = ZZ{Int8}
    Z1 = Z/31
    Z2 = Z/Z(31)
    Z3 = Z/31Z
    Z4 = Z/Ideal(Z(31))

    Z1 == Z2 == Z3 == Z4

It may be noted, that 0R is the zero ideal (containing only the zero element of R) and u*R == R for all unit elements u of R. Currently the expression R / I works only for polynomial rings R.

If there is one generating element of an ideal, a * R, internally a unit multiple of a is stored to achieve a standard form, for example a monic univariate polynomial.

In the case of multiple generating elements, a, b..., an attempt is made to standardize and reduce the stored base. For example if R is a unique factorization domain, then gcd(a, b...) is stored.

Ideals are most useful with multivariate polynomials, when they are best represented by a minimal base - see below.

The class names of all concrete types serve also as constructor names. That means, if R is a class name, then R(a) is an element of R for all a, which are integers or elements of (other) rings, which can be natuarally embedded into R.

For polynomial rings P, the method call monom(P, i) constructs the monic monomials x^i for non-negative integers i. That is extended to multivariate cases monom(P, i, j, ...).

Each algebraic structure corresponds to a parameterized Julia type or struct. For example, to represent ZZ/m, there is

    abstract type Ring{<:RingClass} end

    struct ZZmod{m,T<:Integer} <: Ring{ZZmodRingClass}; val::T; end

The subtypes of RingClass are used as containers for constant type variables. It may be necessary to hold values, which are specific for the algebraic structure, and cannot be stored in as type paramters. That happens for example, if the modulus m in the example above is a BigInt or a polynomial. In other cases, the classes are unused. The user needs not deal with those types as long as he does not define own ring structures.

Access to the type variables is used within the implementation by method gettypevar(::Type{<:Ring}} which provides the RingClass object, when the complete type is known. The ring types typically provide accessor functions to obtain the type specific values, like modulus, order, characteristic, dimension, etc.. Try @code_typed dimension(GF(25)) to see how efficient the generated code is.

For each ring type R the class of polynomials over R is created like P = R[:x] where the symbol :x defines the name of the indeterminate.

Polynomials of this class are obtained by the constructor g = UnivariatePolynomial{R,:x}([1, 2, 3]) or more conveniently by

P = R[:x]
x = monom(P)
g = 3x^2 + 2x + 1

g == P([1, 2, 3])

If R is a finite Field (that means ZZ/p or GaloisField - see below) the following options are available:

Univariate polynomials may be checked by isirreducible(p) for their irreducibility and factor(g) delivers the list of irreducible factors of g. The factorization is also implemented for univariate polynomials over the integers (for example of type ZZ{BigInt}[:x]).

For finite fields, the function irreducible(P, r) delivers the first irreducible polynomial with degree r. All irreducible polynomials of P with degree r are obtained by irreducibles(P, r) which is an iterator. That allows to apply Iterators.filter or find on this list.

While the number of polynomials of degree r is order(R)^r, the subset of irreducibles has order num_irreducibles(P, r).

All finite fields have order p^r where p is a prime number and r >= 1 an integer. It can be represented as a factor ring of univariate polynomials over ZZ/p by an irreducible monic polynomial g of degree r. In short, if g is known, we have (ZZ/p)[:x] / g as a working implementation of a Galois field. For r == 1 this can be identified with ZZ/p (using g(x) = x). The modulus method return the polynomial which is actually used by the implementation.

g can be any monic polynomial of degree r. When constructing the class GF(p^r) = GaloisField{p,r} a brute force search for such polynomials is performed using an efficient method to detect irreducibility. For r > 1 the monom x ∈ (ZZ/p)[:x] / g together with 0 and 1 generates the field by applying addition and multiplication operations. Calculated in GF, we have always g(x) = 0. We restrict the selection of g in order to x let generate the multiplicative subgroup of GF by multiplication. That is possible for all p, r.

Time efficiency of algebraic operations is improved by avoiding the expensive multiplicative calculations in the factor ring and the use of logarithmic tables in the size of p^r. Each element is represented by an integer in 0:p^r-1, which corresponds to a polynomial of degree < r in a canonical manner (for example the number 2p^3 + p + 1 maps uniquley to 2x^3 + x + 1).

We construct a Galois field conveniently by GF(p^r).

julia> p, r = 5, 6; # `p` is prime number, `r` not too big

julia> G = GF(5^6) # GF(5, 6) is also possible
GaloisField{5,6}

julia> g = modulus(G) # the selected irreducible polynomial
α^6 + α + 2° # we use a distinct name for indeterminate

julia> order(G) # `p^r`
15625

julia> x = G[5] # an easy way to obtain the standard monom
{0:0:0:0:1:0%5}

julia> order(x) # `x` generates the multiplicative subgroup
15624

julia> G.(0:p) # the integers are mapped into the field G(5) == 0
6-element Array{GaloisField{5,6},1}:
 {0:0:0:0:0:0%5}
 {0:0:0:0:0:1%5}
 {0:0:0:0:0:2%5}
 {0:0:0:0:0:3%5}
 {0:0:0:0:0:4%5}
 {0:0:0:0:0:0%5}

julia> g(x) # the monom `x` is a root of `g`
{0:0:0:0:0:0%5}

# These are all zeros of g - see also `allzeros`
julia> x.^p.^(0:r-1)
6-element Array{GaloisField{5,6},1}:
 {0:0:0:0:1:0%5}
 {1:0:0:0:0:0%5}
 {1:3:4:2:1:0%5}
 {1:2:4:3:1:0%5}
 {1:1:1:1:1:0%5}
 {1:4:1:4:1:0%5}

julia> g.(x.^p.^(0:r-1)) |> unique
1-element Array{GaloisField{5,6},1}:
 {0:0:0:0:0:0%5}

Matrices and vectors of ring elements are supported. x - A is understood as x * I - A.

The following methods handle vector spaces und subspaces:

For a square matrix, also the following methods exist:

For inv if the element type is P<:Polynomial, it should be widened to Frac(P). det and adjugate work also for matrices over principal ideal domains.

Some example usage:

julia> Z = ZZ / 7
ZZmod{7,Int8}

julia> P = Z[:x,:y]
MultivariatePolynomial{ZZmod{7,Int8},2,Symbol("8693009651133194268"),Int64,Tuple{2}}

julia> x, y = generators(P);

julia> (x + y)^2
x^2 + 2°*x*y + y^2

julia> (x + y)^7
x^7 + y^7

julia> z = (x + y^2) * (x^2 + y)
x^2*y^2 + x^3 + y^3 + x*y

julia> z^2 / (x^2 + y)^2
y^4 + 2°*x*y^2 + x^2

julia> I = [x+2; (x+1)*y];
2-element Array{MultivariatePolynomial{ZZmod{7,Int8},2,Symbol("8693009651133194268"),Int64,Tuple{2}},1}:
 x + 2°
 x*y + y

julia> groebnerbase(I)
2-element Array{MultivariatePolynomial{ZZmod{7,Int8},2,Symbol("8693009651133194268"),Int64,Tuple{2}},1}:
 x + 2°
 y

 julia> P = (ZZ/97)[:x, :y]
MultivariatePolynomial{ZZmod{97,Int8},2,Symbol("8693009651133194268"),Int64,Tuple{2}}

julia> x, y = generators(P);

julia> I = [x^2*y + x*y; x*y^2 + 1]
2-element Array{MultivariatePolynomial{ZZmod{97,Int8},2,Symbol("8693009651133194268"),Int64,Tuple{2}},1}:
 x^2*y + x*y
 x*y^2 + 1°

julia> groebnerbase(I)
2-element Array{MultivariatePolynomial{ZZmod{97,Int8},2,Symbol("8693009651133194268"),Int64,Tuple{2}},1}:
 y^2 + 96°
 x + 1°

# example from [Gröbner Base](https://en.wikipedia.org/wiki/Gr%C3%B6bner_basis)
julia> I = [x^2 - y; x^3 - x]
2-element Array{MultivariatePolynomial{ZZmod{7,Int8},2,Symbol("8693009651133194268"),Int64,Tuple{2}},1}:
 x^2 - y
 x^3 - x

julia> groebnerbase(I)
3-element Array{MultivariatePolynomial{ZZmod{7,Int8},2,Symbol("8693009651133194268"),Int64,Tuple{2}},1}:
 x^2 - y
 x*y - x
 y^2 - y

For F a field with characteristic 0 (for example QQ), it is possible to define a formal power series of a given "precision" over F. You can use them like univariate polynomials. Mind the O(x^n) terms, which indicate, the precision of the expression. We support "relative precision", that means the number of non-zero coefficients is capped by the precision indicator. Lower degree polynomials are exact. The inverse is defined for all power series elements the first coefficient of which is invertible. In the case of QQ, that means all except 0. So this implementation actually supports formal Laurent series.

The compose_inv function delivers the power series expansion for functions f with f(0) == 0 and f'(0) != 0.

P = PowerSeries{10,QQ{BigInt},:x}
PowerSeries{10, QQ{BigInt}, :x}

julia> x = monom(P)
x

julia> 1 / (1 + x)
1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 + x^8 - x^9 + O(x^10)

julia> ex = P(sum((x)^k / factorial(k) for k = 0:13))
1 + x + 1/2*x^2 + 1/6*x^3 + 1/24*x^4 + 1/120*x^5 + 1/720*x^6 + 1/5040*x^7 + 1/40320*x^8 + 1/362880*x^9 + O(x^10)

julia> inv(ex)
1 - x + 1/2*x^2 - 1/6*x^3 + 1/24*x^4 - 1/120*x^5 + 1/720*x^6 - 1/5040*x^7 + 1/40320*x^8 - 1/362880*x^9 + O(x^10)

julia> inv(ex) * ex
1 + O(x^10)

julia> ex / x
x^-1 + 1 + 1/2*x + 1/6*x^2 + 1/24*x^3 + 1/120*x^4 + 1/720*x^5 + 1/5040*x^6 + 1/40320*x^7 + 1/362880*x^8 + O(x^9)

julia> exm1 = P(sum(x^k / factorial(k) for k = 1:11))
x + 1/2*x^2 + 1/6*x^3 + 1/24*x^4 + 1/120*x^5 + 1/720*x^6 + 1/5040*x^7 + 1/40320*x^8 + 1/362880*x^9 + 1/3628800*x^10 + O(x^11)

julia> compose_inv(exm1)
x - 1/2*x^2 + 1/3*x^3 - 1/4*x^4 + 1/5*x^5 - 1/6*x^6 + 1/7*x^7 - 1/8*x^8 + 1/9*x^9 - 1/10*x^10 + O(x^11)

julia> lg = (sum(-(-x)^k / k for k = 1:12))
x - 1/2*x^2 + 1/3*x^3 - 1/4*x^4 + 1/5*x^5 - 1/6*x^6 + 1/7*x^7 - 1/8*x^8 + 1/9*x^9 - 1/10*x^10 + O(x^11)

julia> compose_inv(lg) == exm1
true

julia> using CommuatativeRings

julia> using BenchmarkTools

julia> R = GF(7)
ZZmod{7, Int8}

julia> S = R[:y] # S, y = PolynomialRing(R, "y")
UnivariatePolynomial{ZZmod{7, Int8}, :y}

julia> y = monom(S);

julia> T = S / ( y^3 + 3y + 1); # Factor Ring(S, y^3 + 3y + 1)

julia> U = T[:z]; #U, z = PolynomialRing(T, "z")

julia> z = monom(U);

julia> f = (3y^2 + y + 2)*z^2 + (2*y^2 + 1)*z + 4y + 3;

julia> g = (7y^2 - y + 7)*z^2 + (3y^2 + 1)*z + 2y + 1;

julia> s = f^4;

julia> t = (s + g)^4;

julia> @btime resultant(s, t)
  10.442 ms (259556 allocations: 15.48 MiB)
-y^2 + 3°*y + 4° mod(y^3 + 3°*y + 1°)

This package was inspired by the C++ library CoCoALib, which can be found here: CoCoALib.

The factorization of integer polynomials follows the D. Knuths infamous book "The Art of Computer Programming" chapter 4.6.2.

The signed_resultant_polynomials are from the book "Algorithms and Computation in Mathematics" of Basu, et. al.

The power series algorithms are partially from this wikipedia article Formal Power Series