RationalFunctionFields

RationalFunctionFields.jl is a Julia package for computing with subfield of the rational function field $\mathbb{Q}(x_1, \ldots, x_n)$. The functionality includes algorithms for checking algebraicity and membership, field equality and inclusion, and for finding a simplified set of generators. Most of the algorithms are Monte-Carlo randomized and allow user to provide a bound on the probability of error.

How to install

The package can be installed from the Julia registry by

using Pkg
Pkg.add("RationalFunctionFields")

Basic usage

The package can be loaded with

using RationalFunctionFields

For our introductory example, we consider a subfield of $\mathbb{Q}(x, y)$ generated by three polynomials $x^2 + y^2$, $x^3 + y^3$, and $x^4 + y^4$. We use Nemo for handling polynomials and polynomial ring, so we first create the corresponding polynomial ring.

using Nemo
R, (x, y) = polynomial_ring(QQ, ["x", "y"])

Now we can create the our field as follows

F = RationalFunctionField([x^2 + y^2, x^3 + y^3, x^4 + y^4])

Function field_contains can be used to check if given elements belong to the field with the given probability of correctness (99% in the call below):

containment = field_contains(F, [x, x + y], 0.99)

After this call, containment will be an array of two Booleans false ($x$ does not belong to the field since it is not symmetric) and true ($x + y$ turn out to be an element of the field). Now one may wonder, how $x + y$ can be written in terms of the generators of the field. This can be done using constructive_membership function to which we provide tag_names --- the variables names to be used for the generators $x^2 + y^2$, $x^3 + y^3$, and $x^4 + y^4$.

expr, tag_dict = constructive_membership(F, [x + y], tag_names = ["s2", "s3", "s4"])

Here expr will be an array containing a single element: the expression of $x + y$ in terms of the generators:

(-1//4*s2^5 + 1//4*s2^3*s4 + 1//2*s2^2*s3^2 + 1//2*s2*s4^2 - s3^2*s4)//(s2*s3*s4 - s3^3)

The dictionary tag_dict will give the correspondence between the variables s2, s3, and s4 used in the above formula:

Dict{QQMPolyRingElem, AbstractAlgebra.Generic.FracFieldElem{QQMPolyRingElem}} with 3 entries:
  s3 => x^3 + y^3
  s2 => x^2 + y^2
  s4 => x^4 + y^4

At this point, we may suspect that the field is in fact equal to the field of symmetric functions (generated by $x + y$ and $xy$). This can be checked using the fields_equal function (again, the result will be correct with probability at least 99% since we provide 0.99 as the last argument):

fields_equal(F, RationalFunctionField([x + y, x * y]), 0.99)

Or, the idea to check this equality could have passed by: no problem! We can use function simplified_generating_set which tries to find "a simpler" set of generators for the same field:

simplified_generating_set(F)

And, voilà, it returns $x + y$ and $xy$:

2-element Vector{AbstractAlgebra.Generic.FracFieldElem{QQMPolyRingElem}}:
 x + y
 x*y

Contacts

The package is maintained by Alexander Demin ([email protected]) and Gleb Pogudin ([email protected]).