Universal polynomial

AbstractAlgebra.jl provides a module, implemented in src/generic/UnivPoly.jl for a universal polynomial ring. This is very similar to the multivariate polynomial rings, except that variables can be added to the ring at any time.

To compensate for the fact that the number of variables may change, many of the functions relax their restrictions on exponent vectors. For example, if one creates a polynomial when the ring only has two variables, each exponent vector would consist of two integers. Later, when the ring has more variable, these exponent vectors will still be accepted. The exponent vectors are simply padded out to the full number of variables behind the scenes.

Generic sparse distributed universal multivariable polynomial types

AbstractAlgebra provides a generic universal polynomial type Generic.UnivPoly{T, U} where T is the type of elements of the coefficient ring and U is the type of the elements of the underlying multivariate polynomial ring. Essentially, U can be any type belonging to MPolyRingElem{T}.

Parent objects of such polynomials have type Generic.UniversalPolyRing{T, U}.

Abstract types

AbstractAlgebra also provides abstract types for universal polynomials and their rings. These are UniversalPolyRingElem{T, U} and UniversalPolyRing{T, U} respectively. These in turn belong to Ring.

Polynomial ring constructors

In order to construct universal polynomials in AbstractAlgebra.jl, one must first construct the universal polynomial ring itself. This is unique given a base ring.

The universal polynomial ring over a given base ring R is constructed with one of the following constructor functions.

universal_polynomial_ring(R::Ring; cached::Bool=true, internal_ordering::Symbol=:lex)

julia> S = universal_polynomial_ring(ZZ)
Universal Polynomial Ring over Integers

julia> x = gen(S, :x)
x

julia> y, z = gens(S, [:y, :z])
(y, z)

julia> x*y - z
x*y - z

Adding variables

There are two ways to add variables to a universal polynomial ring S.

gen(S::UniversalPolyRing, var::VarName)
gens(S::UniversalPolyRing, vars::Vector{VarName})

Examples

julia> S = universal_polynomial_ring(ZZ)
Universal Polynomial Ring over Integers

julia> x = gen(S, :x)
x

julia> number_of_generators(S)
1

julia> y, z = gens(S, [:y, :z])
(y, z)

julia> number_of_generators(S)
3

Universal polynomial functionality

The universal polynomial ring behaves exactly like a multivariate polynomial ring with the few differences noted above.

The only functionality not implemented is the ability to do divrem by an ideal of polynomials.

The universal polynomial ring is very useful for doing symbolic manipulation. However, it is important to understand that AbstractAlgebra is not a symbolic system and the performance of the universal polynomial ring will closely match that of a multivariate polynomial ring with the same number of variables.

The disadvantage of this approach to symbolic manipulation is that some manipulations that would be offered by a symbolic system are not available, as variables are not identified by their names alone in AbstractAlgebra, as would be the case symbolically, but by objects.

The most powerful symbolic tools we offer are the generalised evaluation functions, the multivariate coefficient functionality, the ability to change coefficient ring and to map coefficients according to a supplied function and the ability to convert a multivariate which happens to have just one variable into a dense univariate polynomial.

Further facilities may be added in future to ease symbolic manipulations.